My personal favourites are 7x57, 30-06 hands down, and I’ve had just about all ‘notmal’ calibres.
If there was better ammo availability and quality in 7x57 widely available at reasonable prices (Norma are near £60 for 20 here), AND there were more used quality rifles at reasonable prices, I would say start there and never look back!
i ignore my other favourite 30-06 as you mention wanting mild recoil.
finding a really good rifle is most important, as Ed said, any calibre will kill deer on this island...however, from experience, for red deer and especially if you have chances for sika, really, do go above 243. Not only from a knock down effect perspective, but 100g ammo doesn’t always perform well in many 243’s unless barrel length and twist rates are favourable.
secondly, picking that good reliable rifle, then needs to be matched with access to affordable quality ammunition in your area, or you will quickly regret your rifle decision!
It’s not a rifle I would choose again, but I suggested the 308 as there are many quality rifles on the 2nd hand market, affordable quality ammo is easy to find, and bullet weights in the +/-160g range will be very good on any of the deer species you mention.
it was widely said that your first rifle is just a learning rifle from which you pick your next, but might as well make it a clever decision so you don’t feel rushed to replace it.
dont know what your budget is, but Sako and tikka are very good both used and 2nd hand.
If i was on a medium budget, I’d opt for that or buy a new CZ. If I’m a low budget, I’d buy a good but used cz/Brno or Parker hale or sedgemore.
why not 6.5x55? Imho, ammo quality in the 140g class is lacking in most shops, and at anything other than quality 140g, ideally reloaded, terminal performance is poor at best, and for the beginner can leave you with some bad experiences of dead running game. I speak of the 6.5 not with anger or negativity, but with caution