I don't play any of these instruments, but have at least tried them enough to play a scale.
My impression is the basic holes down the pipe are tuned to a diatonic scale. The more holes covered, the lower the tones. You must keep holes covered as you go down the scale.
It seems to me chromatic tones come from fingering "gaps" or on the record from the small double holes at the end of the body, but I don't really know how it works.
Out of the entire extended flute family, you've managed to name three that are about as far apart as they get! However, I'd still say that all woodwind fingerings are more similar than not.
The fife is among the oldest flutes that still get some use, and the fingerings are identical to the tin whistle (a.k.a. pennywhistle, Irish whistle). It has six holes operated by three fingers of each hand. All closed plays a low D, and picking one up at a time from the bottom results in a D major scale. The tone holes are mostly very large, meaning cross fingerings don't work very well in most cases, so most of the "in between" notes have to be done with half-holing, which is sloppy and awkward.
It's a little strange for an instrument to play in D major most naturally, so the history of woodwind fingering evolution can be thought of as a series of improvements to twist that fingering pattern into C major. In order to do that, we need to make F-natural and C-natural easy.
The recorder is the first step in that evolution, and it takes steps towards normalizing C major by extending the instrument to low C (operated by the pinky of the bottom hand--we now always play with the left hand on top, but originally either way was common), and by shrinking the tone holes so that cross fingerings work. But you still get a D major scale with the six main fingers, it's just that F and C are much easier to play. Oboe fingerings evolved alongside the recorder, and so that's the instrument with the most similar fingerings.
The thing that sets the modern flute apart is the Boehm system, which improved on many aspects, but relevant to this discussion, it introduced the automatic F# pad. It's an extra tone hole between the two hands, which plays F# and is automatically closed when any right hand key is pressed, so the old F# fingering becomes an F fingering (and F# is played by leaving the F key open, but closing a lower finger so that the F# hole closes, which is easy). The saxophone and modern clarinet have also adopted this. These instruments, plus the oboe, have also eliminated all cross fingerings and half-holing by giving each chromatic note its own tone hole, operated either by pinky keys or other automatic mechanisms.
So in all, you have (for the instruments you asked about):
C: 1 2 3 | 4 5 6 (pinky) - except fife
C#: 1 2 3 | 4 5 6 (pinky) - except fife
D: 1 2 3 | 4 5 6
D#: 1 2 3 | 4 5 6 (pinky) - flute only
E: 1 2 3 | 4 5
F: 1 2 3 | 4 - on flute
F#: 1 2 3 | 4 - on recorder, fife
G: 1 2 3
G#: 1 2 3 (pinky) - flute only
A: 1 2
B: 1
Bb is a whole 'nother can of worms, as is what happens above B.
The fingerings for almost all woodwind instruments are 'similar' in the sense that covering successive holes gives you a diatonic scale. For the chromatic notes in between, every woodwind instrument is different.
Fairly. I have played them all a bit except fife. (There are also many types of traditional flutes that have slightly different fingerings.) There is also tin whistle which is similar again.
You always basically shorten the tube to raise pitch and they all overblow an octave so most stuff works the same. But there are differences in getting some chromatic notes (of course on orchestral flute you have many extra keys that you don't have on the others so there are a lot of extras).
On recorder you have some oddities like right hand 1/3 to get F natural (I think it is just first finger on tin whistle) but they are largely the same. Of course there are so many varieties that it is a bit harder to answer this any more precisely than that.
This is a supplement to the previous answers.
It's relatively easy to design an instrument so that in the first octave, you just open up an additional hole for each note in order to play the main scale of the instrument. This doesn't mean that this will work in the second octave. The way that the air flow and vibrations create the notes in these instruments is pretty complicated, and various changes have to be made in the fingering for the second octave. A modern flute can make this easier with keys and so on, but on a recorder, you have to do things like cover holes below an open hole, or half-cover holes.
Most recorders are designed to use the English fingering system, and you have to do the same trick in one place in the first octave. Roll your mouse over the notes on the staff on this page to see this. The reason for this is that otherwise, it's too difficult to play some of the other notes in tune. Also, to play recorder well there is a little bit of control you have to exert over your breath and the initial attack of a note in order to get some of the notes precisely in tune. (The initial attack is also important to prevent the recorder from "squeaking". If you listen carefully to someone who picks up a recorder and plays without training, you'll hear a high-pitched little sound at the beginning of the note before it settles into the right pitch.)
Earlier flutes or "traversos" from the period of Baroque and early Classical music usually have only one or two keys at most. In addition to opening and closing non-obvious combinations of holes to get all of the notes in a chromatic scale and to get the upper octave, for some notes one has to do particular things with one's lips and sometimes rotate the flute so that the air hits the mouthpiece at a different angle. This seems like it would be more difficult than playing a modern flute (although a modern flute has a greater range), but when played well, the result is very beautiful, in my opinion.
