I load 7mm Rem Mag for some friends. I own and hunt with 7x57R, (had a 7x57 Mauser), 7mm-08 (3 of them), .280 Rem, and 7x64 (3 of them). Since you have a 6.5x55, might as well jump over the 7x57 and 7mm-08. 7mm Rem Mag is more than needed for any deer, until you get past 300 yards, and 25% more recoil than the 7x64. If you are shooting elk, moose, large bears and wild boar, then 7mm RM is good, but at that point, might as well go to a .30-06 or .300 Win Mag for the heavier bullets constructed for just that.
.280 Rem and 7x64 are plenty for deer, elk, and moose, and very flat shooting, with a lot less recoil than a 7mm RM or .300 magnum. And the rifles are more trim and lighter weight, which is good for stalking. Since 7x64 is easier to buy ammo in the UK, that is my suggestion. You can always load it mildly to 7x57 and 7mm-08 performance. There are some really good bullets in the 150, 154 and 160-gr weights, high BCs and all kinds of constructions.
But if you really want a versatile cartridge which has the big bullets for big game, go .30-06 - 20% less recoil than the .300 Win Mag and will shoot a 200-gr bullet through most critters. If you are a sheep hunter who has the ability to milk the slight ballistic superiority of a 160-gr bullet out of a 7 pound 7x64, you might go that way, but you would also have the ability to do the same job with a 165-gr or 180-gr bullet from a .30-06.