How does one get faster, though? I cannot play very fast, so I end up sitting out even tunes I already know at my session. Sometimes I fear I will always play alone and my music will never sound fun and lively.
Well if you read back you’ll see I wrote just how one gets faster. Maybe I didn’t spell it out letter for letter, but the basics are there my friend. 
-Eric
To acquire speed, practice at a reasonable, comfortable speed. Try to stay as relaxed as possible.
As you get the tune more and more solid, speed it up a bit.
The secret to playing fast is that if it feels fast to you, then you are going faster than you really should. Slow it back down.
When you really have the tune, you’ll be able to go at quite a clip without it feeling fast and frantic either to you or to those listening to you.
–James
I seem stuck at a particular speed, though. Kinda like when god was handing out skill he gave everybody a Porsche but I got a VW bus.
The VW will last longer and looks nicer (not saying much, I know, but at least it doesn’t look like a semi-squashed slug!) Oh yeah, and you’ll get there in the end!
A tune played well and with precision beats a tune played fast and sloppy anyday.
I play half fast, myself…
I’ll second that. Plus I play half slow since you brought it up. 
Alas, human learning is not the smooth upslope we all wish it could be. We humans learn in steps; in learning theory, the flat spots are called plateaus. When we learn something, anything, we make rapid progress for a while then don’t appear to make any progress at all or maybe even fall back (that’s called regression). The thing is, our brain is learning during the plateaus and regressions. This isn’t exactly neurologically correct but think of your brain trying out many different wiring diagrams, trying to find the one that works best. Eventually, the brain gets that step sorted out and we experience another surge of rapid learning followed by another plateau. As we gain experience, the surges and plateaus tend to smooth out but they are still there.
So what does all this have to do with getting faster on the whistle? Keep practicing at a pace you can control, as mentioned above by many folks. One day when you are feeling frisky, step it up a notch. If you lose control, back off; you haven’t crossed the plateau yet. One day, you’ll try to speed it up and, miraculously, your fingers will fluidly and with full control pick up the pace. You’re across that plateau. Enjoy the surge, there’s another plateau ahead.
I’m at best an intermediate whistler but I’ve taught flying and technical courses for almost 40 years. The stairstep learning curve really is how it works.
I’m an intermediate beginner (or may a beginning intermediate) player and I have to say that I completely agree with this. I can’t tell you how many times I’ve been completely discouraged by my playing. Many times I’ve felt like I was getting worse and the the next day, everything just falls into place and I feel like I just jumped 5 steps forward. Don’t give up and don’t get too hard on yourself.
As a retired lecturer, I’m familiar with this and have had a quite striking experience of it learning the flute.
I bought a cheap plastic simple system flute about 2.5 years ago to learn on. Initially I did OK, but then plateau’ed big time. I just seemed to get nowhere. It was finger position that was the problem and that was frustrating as I have been playing tenor recorder for about 15 years and that has similar finger spacing. I effectively gave up and did not touch the flute for about a year and was on the point of getting rid of it when someone gave me a wooden flute. I tried that out and I was away! the fingers somehow just fell into place and I started making progress again.
I have started taking the flute to sessions and have found that even though I tend to play quite slowly at home, I can keep up when out, so a lot of learning has clearly been going on in the background.
There is also a useful thought in this for playing at speed. Practice slowly at home and make sure you have the tune absolutely secure under your fingers and when you are playing with others you will be surprised how quickly you can play. I certainly was.
Geoff
Don’t feel bad. I got the horse and buggy. 