A beginner on a budget doesn't need a rifle that will send him around every gun shop in the country looking ammo, and when he finds it will have to sell his rifle to afford a box.
I'm not so keen on the 243 as it is the "legal minimum," rather than something that is the "stalking ideal" but they are common as muck and you can get ammo everywhere which is one of the key points in picking a rifle and I'd guess they probably kill more deer every year than any other cartridge, maybe more than all the others put together. There are people who say they are the best thing for deer and maybe they are, but that isn't my personal position, but the key is that supplies of ammo and reloading components and so on are readily available. The 243 round was designed for shooting animals such as foxes so maybe you could make it do that job for you? Seems a shame not to put it to the use for which it was designed if it can also save you buying an extra rifle.
A friend recently bought a Parker Hale and spent a little bit of time doing up the stock and it ended up looking great and seemed to shoot pretty well too. Also on this forum there are a number of people who know a lot about them so there is always someone to turn to for advice. I'm no expert but would guess you'd get one for around the £200 mark.
As Brian already mentioned on this thread there is a sako for sale with a scope. That would be an ideal solution as you'd get a rifle basically already set up and if you are not ready to buy right now then I'm sure other, similar, offers will come along.
If you can keep the number of rifles you buy down you can maybe make better use of your cash - the £2k will easily get you 3 rifles, or more, but you are going to want decent glass which is expensive plus do you need bipods, slings, slips, cleaning stuff and most importantly binoculars? All that stuff adds up pretty quickly.