Belly you wouldn't even get to sit the test. It would be picked up by your 170A examiner.
It says in E 1.2 that you must have a PPL or CPL which allows you to fly at night. As part of pre course experence requirments. So I think you are knackard.
And remember as well it needs to be issued so unless you go and sit at Gatwick that will take 3 weeks to come back.
As I said you are goig to have to phone round all the schools who operate out of 24H airports and see if anyone is willing to work at that time of night.
I suspect it will cost you a fortune in landing fee's if you do it in the UK.
Fancy a week in Cyprus?
Griffon
http://www.griffonaviation.com/
You can do the training in any ICAO country although I suspect cyprus is part of JAR
And you will need a MEP course and test cert as well.
I would phone some of the favorite pprune FTO's in Florida you never know one of them maybe able to help.