Replacing the rims is a good thing to do for better breaking and reduction of weight. All you need to know is the size and number of rim holes you'll need to match the existing hubs. Most US Twentys have the BMX 20" size, (aka ISO 406) but European Twentys came with the larger and harder to get ISO 451 size - also (confusingly) often called 20". From the wee pic that goes with your username it *looks* like your bike is a 451 model (the frame and everything else is the same - just the rims and brake calipers are different) As for width - modern 451 sizes tend to be used on higher end folders (Bike Friday) and recumbents so are usually in narrower widths than the 1⅜" size - likely to be 1¼ or 25mm.
FYI your Triumph isn't a 'clone' in the true sense - more a 'badge engineered' Twenty. Raleigh bought up most of the smaller manufacturers in the 20th C and sold Twentys under lots of names - kinda like GM is Chevy, Cadillac, Pontiac etc and puts out cars based on the same major parts.