For the sake of argument, let's say it's Farnborough RADAR, and you're aiming to Popham, but there's a load of crap viz come in from the south, best diversion Odiham.
(You) Farnborough radar, G-CD
(Fbro) G-CD pass your message
(You) G-CD, my visibility en-route is deteriorating and I will shortly be unable to maintain VMC. Request diversion Odiham.
(Fbro) G-CD, roger, standby
(expect a short pause whilst they talk on the landline)
(Fbro) G-CD, I've spoken to Odiham and they can accept you. For Odiham squawk NNNN.
(You) G-CD Squawk NNNN
(Fbro) G-CD, contact Odiham on 119.225, they have your details.
(You) G-CD, Odiham 119.225, thanks for your help.
From there, it's fairly straightforward. As a rule, military airfields like Odiham will just vector you on radar all the way in (making them great safety diversions, apart from the fact that they often don't have AVGAS), whilst civil airfields are more likely to just treat it like a normal arrival.
If you're unsure however, just use plain English, speak clearly, and if necessary route clear of poor conditions first, just stay out of controlled airspace without clearance.
If things are going totally pear-shaped, don't be afraid to ask for a suitable diversion - although in that case many services may transfer you to 121.5 as they have both better locating equipment, and a better database of every tiny landing strip in the UK.
G
Last edited by Genghis the Engineer; 14th Jun 2015 at 22:25.