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.