Is it ever legal in the UK not to have two brakes on a bike? When using a bike on a road in the UK, is it ever legal not to have separate brakes on the front and back wheels?
Yes. A fixed wheel does not need two, it needs only one (on the non-fixed, presumably the front, wheel). 

Two separate and independent bracking systems, one per wheel, are required, but as the CTC say 


  Apart from saying that a brake that
  acts directly on a pneumatic tyre
  shall not be deemed to be in efficient
  working order, these regulations do
  not define how the brakes are operated
  (by hands, feet or maybe even the
  teeth?) or how they work. A lot of
  words are nevertheless devoted to
  wheels that cannot rotate
  independently of the pedals (i.e. no
  freewheel), the effect of which is
  that a fixed wheel drive counts as a
  braking system – on that wheel or
  wheels.


Note also that, when sold, European regulations dictate that it should be left-hand rear, right-hand front.