Voltage across a surge protector plug
Transmogrifying your 


"should there be any voltage?" 


into 


"will there be any voltage?" 


the answer is "There might be, it all depends on circumstances and equipment involved".

I don't know if "a surge protector" tends to be a reasonably specific device in the UK but here in the antipodes at the dawn of time it could mean a wide range of things.

BUT


IF a surge protector consists of a device which conducts when line to line voltage exceeds normal max by a significant margin - such as back to back zeners, a MOV, a transzorb, a gas discharge tube, a neon or similar.
but which is of very high resistance when no voltage is applied 
and IF the surge protector contained X and/or Y capacitors (line to line or line to ground) 


then YES voltage would be very likely to be present, because the capacitor(s) would probably retain some charge if the mains was disconnected with the load  switched off.

If mains is disconnected by opening a switch or pulling out a plug, then line to line voltage and this capacitor voltage could be anything from about +330 to about -330 V as the disconnection timing is not synchronised with mains zero crossing. A capacitor connected across the line or leg to ground capacitors will be left with the mains value at the time of disconnection.  If there is no load present this voltage could remain "for some while".