I was caught out without current tax by 3 volvos, 2 motorbikes and a van with a sodding great antenna and digital cameras. They had some live linkup back to the DVLA database and just ran the plates vs the db and I came up trumps....
I didn't have the V5 (lost in a move) but I had applied for a new one at the time. Told the copper as much, he believed me after craftily saying "ooh, yeah I had that - cost me Ԛ£17 the robbers!" when the correct cost is Ԛ£19 (or something I forget the details now!). I was only a couple of weeks over so the copper didn't even ask me for my other paperwork, told me to go to the DVLA and get taxed before anything nasty happened. The DVLA decided to "settle out of court" for Ԛ£30. I assume that Ԛ£30 went straight into the christmas party fund...
Anyway, long story short...
Get yourself to the local DVLA office with your MOT, Insurance and any other paperwork you can get your hands on. Passport might be a good idea. They have a procedure for taxing cars that don't have a V5 present, its routine for them. The pain in the arse is that they only open strict local government hours, ie 9-5 with an hour off for lunch so you need to either nip out from work or get an hour off.
DVLA local offices
From what I can remember/read on the site.. they will tax the car if you fill out a
V62 and use that instead of the V5 over the counter.
Just the cheer you up the fine is up to Ԛ£1000 according to
This