Your choice in all three calibres has merit. Personally, I would discount the .270 as there is not the choice of bullet weights in this country compared with the other two calibres. The .308 is a great choice, with widely available factory ammunition and the case fits into a short-actioned rifle, making a compact package for stalking at home and abroad.
The bullet weights range from 90-grain up to 240-grain, and one weighing 150- grain or 160-grain is a good all-rounder, especially for red deer and fallow, with the heavier 180-grain or 200-grain ideal for rutting woodland stags or wild boar.
Anything the .308 can do, the 30-06 can do better, but it burns more powder and has more recoil. It also has a long action to cycle, and is usually heavier than the .308. The 30-06 comes into its own when shooting the heavier bullet weights from 180-grain upwards, offering 150fps velocity more than the .308.
My personal choice would be the .308 with 165-grain bullets at 2,750fps for all British deerstalking.