ok im guessing you would want to go with a late model C15 or B25 BSA? they are pretty slow but you can do a bit of work to them. ive seen big bore kits for C15 non ebay. i really think you would need a good ignition, larger carby, something more modern like a mikuni, a good larger diameter free flow exhaust and some serious porting in the head and bigger valves. theres also mods to do to the bottom end to convert the timing side main to a bearing instead of a bush in the C15. a fair few dollars would need to be spent to make it enjoyable to ride and at least feel quick and responsive because as thay are std they are a bit of a slug.
so my thoughts are if you dont want or need to be higly competitive and dont mind something thats going to be a little slower than go for it.
have you ridden stock british 4 stroke 250?
as i can see from that triumph, i bet that is no where near std inside.