Henry,
Don't even bother filling in the variation form. I can confirm from personal experience that the immediate answer will be a refusal. In addition to which, the military and thereafter the NRA recently passed specific rules concerning firearms to which the 338's is included that restrict their usage on ranges.
I had so much hassle with my 338 Lapua when moving from an English constabulary to a Scottish that i ended up just selling the rifle. In the end, i'm glad i did sell it then because a few friends have been forced to sell theirs, it's now a buyers market and they've lost alot of money.
You'll cry when you buy the ammo - even if you intend to reload for it. Over £100 for 100 good lapua cases, double the quantity of powder = £70 a bottle!
Question - what the hell will you shoot with it? I used to use one in order to try and buck the wind at long ranges - ranges around 1000yds. For example, my most consistent load was a 225gr bullet with 91grs of vhit160 - this threw the bullet out at approx 2700fps. That's around the same speed as a .243 round with less than half the bullet weight. I'd imagine the meat damage to be excessive and any stalker/keeper would likely boot your arse!
If it's just a 'heavy hitter' you want then why not consider the .300 Win Mag? It has similar long range capabilities, 3/4 of the bullet weight and only marginally more expensive to reload than a 308 / 30-06.
If you're determined to get an 'open ticket' with a .338 then i suggest you tell plod that you've found a colony of Grizzly Bears on your patch