Having shot many roe and red with a .243, 6.5x55 and 30-06 as well as 7x57...the .243 even if the twist does allow 100g bullets, is marginal for red deer, and even worse for sika..I can't comment on fallow. On calves or small yearlings, fine..but mature hinds and stags, especially those where you don't want a long follow up near woodland, etc., it's just not good enough IMHO. I have commented extensively on this subject before and have firm opinions on it based on evidence from years of usage. 125kg dressed stags, even with a mixture of H/L, High Shoulder, etc. shots, with noslers, norma's, etc. from 80yds to 220yds have all required 2-3 rounds to firmly anchor them at times, not always, but far far too often.
I'd never ever ever use a .243 on a rutting red stag with a neck shot through a thick mane where just finding the vertebrae is hard enough, that's (neck shooting) not even a subject to back up the .243 on reds.
The 6.5x55 I've used for around 5 years and probably culled somewhere between 100 and 200 reds with, along with a good number of roe. - The 6.5 if reloaded and used with a max weight 140g expanding projectile that's not too stout, will perform adequately on all UK game. factory ammo can be aenemic and heave for calibre bullets do not expand sufficiently or carry enough velocity to exert good levels of shock and terminal performance. Seen too many roe on dead runs, same with reds..the 6.5x55 kills just fine if the right bullets are used at the right speeds and weights, so if opting for a 6.5, pick your bullet carefully such as a classic SP in 130-140g for roe and nosler partition for reds in 140g.
IMHO, I'd pick a 7-08 if I was you. better all round, only superceeded by the 7x57 of course.