It's not the best time to be in London - or Scotland, even - as many places are shut for the holidays. However, I note from your profile that you might enjoy one or other of the Christmas services at St Paul's, or Westminster Abbey, perhaps? The downside being that you'd have to walk there (maybe) and back for sure. Although if you're confident enough to ride a 'boris bike' that might help a bit. (Check out the churches and also the bikes, on google.)
Scotland - Edinburgh for Hogmany (New Year's eve) is a must. Except that you might have problems finding somewhere to stay at this late date.
All public transport stops in the UK on Christmas day, and coaches don't run on New Year's day either. Museums are shut on Christmas and Boxing days. Most restaurants are closed on Christmas day.