Hi,
am not an authority, so feel free to check further but I think that you will find that in the UK a bike (or scooter) with an ic engine is legally a motorcycle so you need a licence, insurance and to pay vehicle excise duty. At the lower end you can keep it to less than or 50cc, max speed 30mph and a few other criteria and register it as a moped in which case it can be ridden at 16. The rider will need to pass a CBT test too. The bike will also need an annual MOT test.
THis is why things like motorised scooters are confiscated by the cops if used on public road/pavements.
Also, any modified bike will need to pass an SVA (Single Vehicle Approval) test at the local goods vehicle testing station before it can be licenced. All in all a minefield of legislation to keep what would only be a few bikes off the road.
There's a good link here that may be of interest:
http://www.users.globalnet.co.uk/~pattle/nacc/arc0497.htm
Now you could convert a bike to electric, that's legal for over 14's. No tax insurance or test but you are limited to 200 watts rated power (250 watts in more normal in Europe and I think you'd talk your way out of that if pressed). However that isn't a lot of power.
However, an avenue that you might want to explore is something I am contemplating; returning a defunct Autocycle to the road. I have aquired a frame for a 1950's New Hudson Autocycle off Ebay for the princely sum of £15. It will be rebuilt using mainly more modern parts such as brakes and wheels for safety and reliability but will in essence be a restoration. The vehicle identity travels with the frame so it should be possible to get an age related plate for it. The bike should be re-registerable with the DVLA and so avoids the complication of the SVA test that goes with a new build. Hopefully it also classes as an historic vehicle so qualifies for free road tax (VED). One complication is that I would need to get it classed a s moped - originally it would have had a 100cc Villiers engine which makes it a motorcycle. An annual MOT test and insurance would still apply. A moped licence would be ok - I'm lucky as I passed my test in the days when they gave you right that with a car licence, today you have to do a CBT and pass a test.
The idea sprung from here:
http://www.users.globalnet.co.uk/~pattle/nacc/arc0548.htm
I hope that this is useful. As I said, I don't regard myself as an authority on this subject but like you am interested in persuing the idea and would be pleased to know what anyone else thinks.
How old are you? This has quite a bearing.
regards
Colin