The Raleigh Twenty was supplied with two different sizes of wheels.
The Decimal 20" x 1.75" size (406mm rims) and the Fractional 20" x 1-3/8" size (451mm rims).
It seems clear that the Twenty was originally designed with the 451mm rim size in mind. The brakes on Twenty's supplied with the 451mm wheels were similar or identical to the ones used on the various Sport-Roadster and 10-speed bicycle models made at the same time.
It seems that the 406mm wheels were mostly found on export models. They required a very long reach rear brake. Weinmann 1080 brakes were used on the rear on some models. The usual wheels were made with 28-spokes, although some were supplied with 36-spoke wheels.
As far as I am aware, all Twenty's were originally supplied with steel rims. These were usually pretty good quality items at the time, and unless they've severely corroded, many remain serviceable up to and over 40 years of age. They are very sturdy, but they are rather heavy and do suffer from poor braking quality in dry conditions, and basically don't work in the wet.
According to Chuck Glider, The original steel rims each weigh 550g. He replaced these with Sun-Ringle ICI-1 alloy rims, which are 320g. That is a 230g just replacing the rim! 460 grams shaved off the weight of the bicycle, improving acceleration, and braking at the same time.
Aluminium Alloy rims will solve the issues of weight and poor braking. The Sun CR-18 rim is available both in 406mm and 451mm, but supply is difficult in some places. 406mm rims are much more widely available, being the common size for BMX bicycles, but are not usually available in 28-spoke (36 spoke is very common). 451mm rims are not widely available either, but what is made, is for the Recumbent bicycle market and very good quality.
I built a new wheel for "RADAR" using Araya VP-20 alloy rims (road find, would you believe?) stainless steel spokes (ebay) and a Sturmey-Archer SRC-3 (II) hub. (See this blog posting). It is not difficult to do, and I can vouch that 3-cross lacing is quite suitable for hi-flange hubs in the small 406mm rim size.
406 v. 451
Suitable Rims in the 406mm and 451mm
No doubt there are more rims available that I have not yet come across. I would venture to say that they are most probably intended for either BMX or Recumbent/HPV machines, but would probably prove suitable for use with the Raleigh Twenty.
Tyres in the 406mm size are so common that it is not worth listing them. Please note that the widest tyre that will fit in the Raleigh Twenty's front forks is 20" x 1.75" (47-406mm). You will need to replace the front forks if you want to run 20" x 2.00" (51-406mm) tyres or wider.
Tyres in the 451mm size are more difficult to obtain. There are quite a few in the 28-451 size, and are used in recumbent and BMX racing bicycles. The wider 37-451 size (traditional 20" x 1-3/8") is difficult to obtain, but the Primo Comet tyre is made in 37-451 and 28-451 and is probably one of the best tyres in this size. (It's also made in 406mm). There are various cheaper tyres made in 37-451, but they are most probably intended for wheelchairs.
Schwalbe only make a slick tyre in 28-451 called the Durano HS 399. They also make a couple of 25-451 tyres, but at this width I would suggest that they would only be suitable for racing or juvenile use. There used to be the HS 110 available in 37-451, but it is no longer listed on their website, although a few retailers of recumbent bicycles seem to have them in stock. They also produce the Mow Joe HS 371 in 37-451, which is a BMX racing tyre designed for non-paved surfaces.