Edit: I misread your post. I thought you were wanting to be an expert whistle player within a couple months. Anyway, I added some to the end of the post that addresses what you really asked, with what I said in my answer in mind.
The whistle is an easy instrument to learn, but there are challenges involved, which may limit what you can do, depending on experience and knowledge. There always seems to be more to learn, if you're willing to learn it, just like a lot of things.
I began teaching myself to play the whistle about sixteen years ago. I already knew how playing the piano worked, but I didn't have much experience with other instruments at the time. So, I'll share my insights from that perspective.
To answer your question, I think a lot of it really depends on what other instruments you've played, how well you played them, how much music theory you're willing to apply, what you consider an expert, and how well you already know how to do ornamentation as it pertains to Celtic music. It also depends on what songs you want to play, and what key you want to play it in, with what whistle.
Playing accidentals well takes a lot of practice, if you want to do that (and if you want to be an all-around whistle expert by my definition, you'll want to know how to do that well—not just okay). I'd say it takes at least two to six years to learn how to play accidentals really well, unless you're a natural (covering half holes, in my experience, usually gives a much clearer, stronger sound than the special fingerings a lot of people prefer).
Ornamentation, or playing Celtic style, is much harder to learn than the whistle itself, in my experience. In fact, I haven't learned how to do that well, yet. Watching close-up videos of the fingering in slow motion seems like it could help, but all I had to go by when I was trying to learn was text, and audio and video that went too fast and was too far away. Nevertheless, that didn't deter me from playing, and I can attest that Celtic-style isn't the only way to play a whistle. I think it might be easier to learn Celtic ornamentation on a melodica (but easier to play ultra fast on a whistle if you know how, I'm thinking).
Whatever the case, although I don't know how long it takes to learn ornamentation on a whistle, I wouldn't call it easy to learn from reading a book.
If you're familiar with recorder fingering, you'll probably have a harder time learning whistle fingering than if you start from scratch, because a lot of people who play the recorder disbelieve that it isn't fingered like a recorder on the upper notes of the first octave. It's much simpler. Don't complicate it by remembering how a recorder is fingered. The basic knowledge of how a whistle is fingered can be learned very quickly (like ten seconds, if you're familiar with other instruments and fingering charts). What's not as obvious is all you have to do to enter the second octave is blow harder (although there is a special, but simple, fingering for the first note of the second octave). Same for the third octave (if you can play notes on it; you can usually get up to three notes into the third octave, but it really depends on the whistle, and you).
Learning a whistle at an expert level may pose challenges if you're always around other people (because the second octave is quite loud and may disturb them if they're not intentionally trying to listen to your performance).
Learning how to get clear notes in the second octave can be a challenge at first, but it gets a whole lot easier with practice. How easy this is, and how easy accidentals are, also depends on the whistle (in addition to your skill, and your skill with the particular whistle).
If you're familiar with playing the piano, you'll also need to realize that you don't use the tips of your fingers. You lay them flat, so the pads of your fingers cover the holes. Learning to cover the holes properly is one of the biggest obstacles for beginners, especially as they can appear covered and not be. This is a much bigger challenge with low whistles. You don't have keys to cover the holes as with a flute (so it takes practice to do it). Not everyone has the same style and shape of fingers (so, this may be easier or different for some people than for others). I can't say how long it takes the average person, but I have fairly prominent pads on my fingers (so, I probably had less trouble than a lot of people might).
Tonguing also takes practice. I'm not sure how much tonguing a whistle is like tonguing other instruments, but there is more than one way to tongue, and each may take a while to learn to do well.
Most people don't know this, but you can play a half-step lower than the regular lowest note of the whistle. I recommend learning it, since it increases the number of songs you can play in a desired key quite a bit. You might be surprised how many songs have that one note that's just barely too low for the whistle's regular range. To play this lower note, cover all the holes, and half of the very end of your whistle. With a small whistle, you can use your pinky to cover it. With a low whistle, you can cover part of the end by leaning it against your shirt or something. It totally doesn't seem like something you would actually use in a timely fashion in a song, at first, but you can, with practice.
Getting used to transposing songs to the key of your whistle takes some practice, but it's fun, and with a two octave range, you'll probably want to learn as much as you can about transposing, if you read sheet music. Fortunately, it's much easier to do than the same transpositions on a piano. All you have to do is finger your whistle like it's a whistle in the key of the song you're playing, because if your whistle is in D and the song is Bb, then D and Bb are both tonic (or Do). So, you don't have to play accidentals to change the key to the key of your whistle. You just pretend your D whistle is a Bb whistle, and it'll play it in D (it'll play it in the key of your whistle). So, getting quick at recognizing which key a song is written in is very helpful. If you enjoy playing songs where you'll need to do such as this, I recommend just memorizing all the key signatures in the major scale, because then you won't have to think about it. Playing the whistle is good incentive to learn some basic music theory (because it vastly increases what you can do).
You may also want to learn how transpose in ways that require you to cover half-holes, too. Normally, the less half-holes or accidentals the song requires the better, but I recommend getting used to it.
If you just want to concentrate on playing the music well, without knowing everything there is to know about playing a whistle, you can do that. I'm sure a lot of people do. I suppose you can be an expert without being an expert in every aspect of whistle-playing.
Some people might not believe you can really get good at songs with lots of accidentals on a whistle, but it is indeed possible, if you're determined.
Vibrato on a whistle, which you may never use in Celtic music, can be accomplished in at least two ways, both of which require some practice and skill. A lot of people wonder how I can do vibrato on a whistle without shaking it, but it's a lot like singing, with the air involved. Just know that you can do it on a whistle, and it's easier than with singing, in my opinion.
In short, I wouldn't count on becoming an expert in a couple months, but you can learn a lot of stuff and maybe even get pretty good, with previous experience in musical instruments. Probably what takes the most time is getting it to sound good. The longer and more you play, the better it'll probably sound. Previous wind instrument experience should help a lot here. If you're new to wind instruments, it may take a lot longer.
Again, playing the whistle isn't one skill. It's many. However, you can probably still become a world-class musician without being an expert in every last skill I mentioned, but the skills will broaden your horizons. I imagine you could accomplish your goal for video game music, but it's hard to guarantee. Some beginners sound pretty good right off the bat (including a saxophone player I know who tried it). It depends. You probably should try it, though. Whistles can be inexpensive and still very playable.
