Our earliest textual mentions of Britain (writen by a Greek navigator named Pytheas, who claimed to have sailed north past Britain) describe it as an island, and this knowledge wasn't forgotten during antiquity or the early middle ages.

Early Roman descriptions of Britain make a big deal about its geography. From a Mediterranean perspective, the Pillars of Hercules - the Straights of Gibraltar - were the (super)natural limits past which boats weren't meant to go. People did go past them, of course (we have evidence that Phoenician traders went south around Africa, and there's Pytheas' account of sailing north), but sailing into the Atlantic was a big deal. The Roman emperors who conquered, and later traveled to, Britain capitalized on the fact that it was an island on the edge of the map; there was political capital in executing such a bold act of navigation, and in having the power to bring an island outside the habitable world into the embrace of civilization. Emperor Constans, in the mid 4th century, made a surprise visit to Britain in the middle of the winter (when the North Sea is rough) just for the hell of it, to show he wasn't afraid of the distant ocean and that he could go to an island at the end of the earth whenever he wanted to. He struck a bronze medalion when he got back to Rome which showed himself in a boat, pointing a spear down at the ocean and carrying standards that suggests he had just won a military victory over the distant ocean.

In the early middle ages, British maps like this Mappa Mundi put Britain on the very edge of things (lower left corner in the map linked), stressing its separate and insular position. These maps are centred on Jerusalem, with Britain representing the furthest reach of the gospel - a separated (yet special) island, literally in one of the corners of the earth. (Also note the shape of Britain - it's distorted by the map's projection, but clearly based on detailed knowledge of the coastline; which is unsurprising, as transport / trade by sea was far less expensive than transport by land, so navigating the coast was common.)

Did people in Britain recognize themselves to be an island, and on the edge of things, before the Roman empire brought them into the Mediterranean economic zone? That's difficult to say. There are periods in the island's history during which oceanic trade seems to have been a little less common; perhaps knowledge of the outside world ebbed and flowed during these periods, and people's sense of their place in the world sometimes contracted so that they thought that Britain was all there was. People in prehistory were often very mobil, though, so I suspect - but can't prove - that consciousness of Britain's geography goes back far into prehistory (perhaps all the way back to ~8000 BP, when Doggerland - the strip of low land that connected Britain to the rest of the Continent - was flooded by the North Sea and Britain and its inhabitants became isolated from the rest of Europe, save by boat).