With only 20k to play with you are not going to build anything I don't think. You will need circa £40k to build even the humblest of machines by the time you have included the dreaded Vat and fitted basic instruments and got it permitted etc.
So I would seriously consider some of the excellent used suggestions Genghis has made.
There are obviously cheaper aircraft out there to be had but my advice would be stick to LAA machines they will be so much lighter on your pocket for maintenance than certified aircraft.
Also look and consider a Jodel, Condor or Emeraude, quite old machines now but dependant on your level of maintenance expertise they would be excellent choices.