This is basically a re-badged Raleigh Twenty isn't it?
Have a look at your current wheels. They will be one of two sizes, 20 x 1 3/8" (or 451 x 37) or 20 x 1.X (405 x ??). The 20 x 1 3/8 is more likely and is a traditional British size, also used on wheelchairs. Some of these bikes were factory fitted with the 20 x 1.x BMX size, although I think they were mostly export models.
I would have a preference for the slightly larger, narrower 20 x 1 3/8" wheel but you may have a problem. There is a very limited selection of rims in that size. SJS stock them, or at least used to, but only in steel. Not good for braking but I wouldn't image the weight penalty would be too big on a small wheel size. Continental and Schwalbe do decent tyres in this size so there should be no problem there. The cheap as chips Raleigh Record tyres in this size aren't that bad either, just a little heavy but seemingly fairly puncture resistant. I have the white wall Records on mine.
The BMX size would be much easier to get rims for but I wouldn't go too wide as the front forks are quite narrow on these. (Also watch the front hub as the fork is made to Raleigh dimensions and is narrower than most.) If you are converting from 20 x 1 3/8" to 20 x 1.??, you will most probably need longer reach brakes. If I wanted to upgrade mine, I would stick a Weinmann centre pull on the front anyway as the standard calipers are just so-so. Also note that the BMX wheel is physically smaller so it will drop the BB, maybe not a good idea on a fixed gear bike.
If it is the re-badged Raleigh, there is also the issue of the 26tpi bottom bracket cups and longer than usual bottom bracket axle and the weird headset with the nylon bush. A standard Raleigh threaded headset can be fitted but are difficult to find. If there is wear in the nylon bush, just pack it out with some sort of shim and grease.
