Because of arthritis, the thumb joints on both hands are gone, after playing acoustic guitar with a pic for 40 years in Church folk groups. I can't squeeze the neck with my left hand and can't hold the pic with my right hand - actually I can do it, but much pain follows. What kind of a guitar can I play without thumbs?
Consider switching to the ukulele. With only four strings, your left thumb is not needed to hold down strings, and you might find the lower tension and smaller neck of the uke easier to hold than a guitar. A pick is not needed with a uke, and many people strum with their index finger, so your right thumb is not necessary, either.
You could transition to lap steel, or dobro; using finger picks on your right hand and a slide on one of your left fingers shouldn't require use of your thumbs.
Maybe autoharp? -- not quite a guitar but still a strummed stringed instrument. From what I can tell, it should be playable without using the thumbs too much.
To add to the answer that mentions lap steel, David Gilmour can be seen playing lap steel here: http://en.wikipedia.org/wiki/Lap_steel_guitar.
Jeff Healey (before his untimely death aged only 41) was a rock/blues/jazz guitarist who held his guitar like a lap steel guitar. Because he largely taught himself from the age of 3, and was blind, he wasn't influenced by seeing the way others hold the guitar. If you google, you can even see photos of him standing up playing the guitar overhand, which looks decidedly awkward.
Playing the guitar like this will obviously take some re-learning (as it would if you were to play lap steel) but there's no reason why only country music should be played with the guitar in this position.
Granted, this will be easier to do on an electric than an acoustic, but if you have a decent acoustic with a low action it shouldn't make much difference.
After getting over crippling tendonitis in both thumbs, I gave up guitar in favor of bass guitar. I can't explain the reason that bass is less painful, but it definitely is. I also had an injury to my right thumb that makes gripping a pick difficult. With bass, I can use my fingers, or I use a Herco pick with loop on it, similar to the thumb pick that a banjo player uses. That keeps me from dropping my pick. Herco picks would probably be helpful with other instruments as well.
Assuming your guitar has a decent action, there's no need to squeeze the neck with your thumb to play.By holding the body with your other elbow/arm, your chords and single notes can be held by counteracting this tension pulling backwards with your fretting fingers. It will feel rather strange after all those years, but it does work. For single note stuff, watch BB King or such like. His 'butterfly' vibrato works because his fretting hand is not squeezing the neck. If it did, it wouldn't work. Open chords are possibly easier than barre chords, but you should, with practice and perseverance, be able to change how you play, not involving the fretting hand thumb as much, or at all. I'm having to do similar, as my thumb is starting to give me gyp.Good luck. Adaptation is far better than no more guitar playing !
As far as the other hand is concerned, a pick that fits over a thumb or finger might be your salvation. Depends rather on whether you pluck or strum.
You could learn to play a tap guitar. Though not exactly a new kind of guitar, the most common tap guitars are low-action electric guitars. Adam Fulara, for example, often plays a double-neck tap guitar. He does not play the instrument with his thumbs, but his thumbs still support the guitar while he bangs the guitar with his other fingers. Though Fulara shows what can be done, I suppose his choice of music would be physically punishing for anyone, including himself.
Another option is the Chapman Stick, which does not seem to be as demanding on thumbs as the double-neck seen above. As you can see in this video, Culbertson's right thumb seems to be completely free. His left thumb still supports the guitar.
Here are two examples where the players seem to be resting their instrument on their shoulder part of the time. When Renat Bikchurin plays his "retar" and it seems to be clamped between his legs, and leaning against his shoulder. His thumbs seem to be just guiding his palms. I suppose you could use this technique with an ordinary tap guitar. And Clubertson seems to do the same thing with his Chapman Stick.
And then there is the "Kelstone" which is positioned like a keyboard and played like a guitar, among other things.
I can play my electric guitar okay thumbless.
Left thumb: You don't really need your left thumb to do most things on guitar, although it generally makes things a little easier and less awkward.
The biggest limitation I find without a thumb is that barre chords are more difficult—with my instrument anchored at three points (body, leg, arm), using arm weight I can play a lot of basic barre chords pretty easily, especially higher up on the neck, but you won't catch me playing an F chord without my thumb! (Well, with a few moments to prepare I can just barely do it, but I wouldn't be able to switch to it in a song without flubbing it.)
Still, I can play melodies and simple barres (especially further up the neck) fine without using my thumb. I find that I can do vibrato and bends fine without my thumb as long as I'm bending toward the ground (toward the higher strings), but I can't seem to bend the other way without using my thumb, so that's another limitation. My guitar has a Floyd Rose, though, so I can do vibrato just fine on the high notes anyway!
Besides that, you can play a lot of music with just open chords, using a capo when you want to change keys, so if you want to play more folk-type music that's an option, too.
Right thumb: Grow out your nails (very slightly!) and alternate picking with your index and middle fingers like a classical guitarist or a bassist. If you can still use the base of your thumb (the area near the wrist) for muting lower strings, that could help a lot. You could also get fingerpicks if using your nails is a problem, or just use the tip of your finger instead.
You can also strum with the back of your index finger nail, or even strum up and down with just your index finger. I can play some folk music just playing open chords and strumming with that single finger, not worrying about my thumbs at all.
I've been sitting around playing thumbless for the last hour, ever since I read this question. I feel pretty confident that if I lost my thumbs tomorrow, I could keep playing like this!
Classical guitar should be playable quite well without thumbs. The elevated neck and firm sitting position is in part employed so you can put more pressure from the arm on the fretboard: the thumb isn't really used for pressure at all, only for orientation. And since you don't use a pick, there's no problem holding one with the right hand, though obviously a lot of finger techniques also rely on the thumb – but you can certainly do melodies and strumming, and for arpeggios there should also be thumb-less solutions when you search them.
Frank, if your problem is osteo-arthritis, my sincere sympathies. I've got this and am still trying to figure out how to work around it. If you have a good physiotherapist (or physical therapist) who specialises in hands, they may be able to concoct a splint that immobilises the relevant joint/joints, to divert the pressure to a joint that is less affected. For a right hand player, the left hand normally uses four positions (classical players can turn away now): the left thumb upright ; the left thumb hung over the fretboard on the top 2 or 3 strings (Hendrix-style) ; the left thumb parallel with the neck, pointing to the headstock ; the tip of the left thumb pushing into the back of the guitar's neck. See how often you use each of these and which is easiest (if any). As many people above have noted, many people grip more than is necessary, and you need to practise without using the left-hand thumb at all, using finger weight. The problem is that you will then push the neck back with your fingers -- now you realise that for all these years, your thumb has been resisting that pressure. Some people have suggested playing a lap steel ; another way of changing posture is to hold your guitar upright, like a double bass player -- now you shoulder is resisting the finger pressure and you can ignore the thumb. This may help your right hand problems by changing the wrist angle and bringing in your index and second fingers like a double bassist. Alternate tunings are worth a go, since you will probably find your left-hand spread is now decreased, so you will need to compensate by moving your forearm or wrist more. Open chord tunings are not necessarily the best answer, since they are really good for barre chords -- and we probably aren't doing those any more. So work on alternative 4 string tunings that allow a switch between minor and major, adding 6th, 7th etc, with 1 or 2 fingers only. Bottom 4 strings for me (DGBE) I am trying as CFBbEb, so that I can still do chromatic runs, and have even tried D F# A# C# to bring the notes closer together. I would recommend a string-winder, even for tuning, and using the whammy bar for vibrato rather than left hand. Protect your hands at all times, rest and splint if it helps, use warm water for soaking, massage the muscles under the webbing between thumb and forefinger, use any kind of medicinal rub that helps, and take tablets such as ibuprofen if possible. Wear gloves and don't knock the joints. Your physician will recommend whether to have cortisone injections or not (hopefully not, as the consensus is they are painful and don't help long-term), and be aware that oral prednisone can reduce bone density. Good luck, I'm still trying!
What if instead of switching guitars you switch hands??? You could try some kind of attachment to your hand to make the grappling and picking easier. Putting something between your index and your thumb could work and you could get used to it faster than trying to completely change your technique. I know this is not what you asked for but was the best i could come with.
I have the same problem in both thumbs, but you shouldn't be using pressure to grip the neck with the left thumb. It's not needed at all. Many classical players just touch the neck with the left thumb. As for the right thumb, best thing is to use something like the large Clayton rounded triangle picks. Jim Dunlop triangles also good, but they must be as large as the Claytons. Use classic pick grip with it placed on the SIDE of the forefinger, push the pick back to rest on the first inside joint of your thumb, so the top of the thumb is sticking out, and much of the pick is stabilised by the top of the triangle resting on the inside of the forefinger nearest the palm. Then you can relax, and not much grip is needed with the thumb. You can even play with an inflamed joint like this. Also, don't get dehydrated!