I come from a formal music background (not only classical, but some of that, certainly), and that shows in my practicing, but I think there’s something to it even for just playing trad. So here’s some of the things that are a little beyond the “pick up the whistle and work on tunes” sort of practicing.
First, schedule. You don’t want practicing to be so in the way it’s a chore, but you don’t want to leave it squeezed in around everything else, either. The amount of time you practice is a function of how you learn, how well you play, and how much spare time you have, but regularity will help things come together quicker than they otherwise would.
Second, warm up. Don’t jump right into playing tunes, especially at speed. Take the time to get used to the instrument again. Working on tone is a good way to do this, and you can squeeze working on breathing in as well. (How many people concentrate on their tone at all? It’s not just the instrument making a particular tone! Especially with the low whistle or flute.) I tend to start out with long tones and scales and arpeggios, but that’s probably a bit of my classical background coming through; starting with airs would work similarly, but concentrate on clean tone and breathing, rather than listening to the melody on the outside. (More about that later.)
Repeat. It’s how our brains work; a sure way to teach your hands and mouth and lungs and brain to work together without having to consciously coordinate movement is to do it over and over. Don’t jump along to the next tune when this one gets passable; certainly that’s a sign that you can start on something else, but get a tune perfect. Come back to old “simple” tunes later on and see what you can do with them.
Mix it up. Too much repetition gets stale, and people do reach plateaus, so you need to learn to recognize when you’re not progressing because you’re not progressing, and leave a bit alone for a while. Work on something else; pick up the low whistle if you’re stuck on the high, or vice versa. Back when I was playing the bass for a living, I got a little burned out gearing up for one concert and put the bass down for a week after it was over; when I returned to it, the solo work I’d been working on seemed so much easier.
Listen critically. With catchy tunes like we tend to play, it’s easy to start listening from the outside, like an audience member would, at the shape of the tune or the ornaments you played before – or, at the flub four bars back. Instead, get into the sound, feel it as though you’re producing it (you are!) rather than listening to it come out. This is always disappointing when you start, but it’s worth it in the long run.
Work from the end. Ever notice that the weakest part of a tune you’re learning always seems to be the end? That’s because when we make a mistake, we go back to the beginning of a section or tune. Try learning a tune from the end sometime. First the last two bars, then the last four, then the last eight, and so on, until you’ve got the whole tune down.
Move. Change the way you play; if you usually practice standing, try sitting for a while, or vice versa. If you usually play in the bathroom or garage to get that rich sound, go to a carpet-filled rec room with an uncomfortably dead sound. If you usually play in a dead room, go to a live one. (I’d recommend usually playing in a dead room; they’re considerably more honest.)
Have downtime. Sometimes, even when you’re working on getting a set down for a gig this weekend!, it’s good to take a bit of time at the end of a practice session (they have ends, since you schedule!) and just play what you want to play. My usual playing is all trad, so when I want to wind down I’ll try playing jazz standards or passages from Baroque music and so on.
Get feedback. The best feedback is a teacher, even if it’s a once-every-six-months lesson. It doesn’t necessarily have to be a whistle teacher; obviously a flute or UP player could help a lot, but a fiddler could help too because they know how it’s sppposed to sound. It doesn’t have to be a paid teacher/student deal either; sit down with a friend that plays and go over each other’s playing. And record yourself and keep the recordings! Go over them right away to hear how you sound out in front, a day or a week later to hear things you didn’t know you played, and months later to judge how you’ve improved (and if you still have some of the same problems to work on!).
Lastly, have fun and work hard. It’s all about balance between the two. Practicing isn’t always fun because there’ll be tough sections to get through and bad days and so on, but never let it all be miserable work, either. 
(Wow, that was long. Sorry about that.)