The Isle of Dogs isn’t actually an island, although it was once an oxbow island. It’s a very grotty part of East London (which I kinda like nevertheless).
The Isle of Wight isn’t remote or rural at all. It’s a seaside resort on the South coast, quite picturesque, not particularly windswept.
The Isle of Man has a bizarre feudal society. Some of its laws are different to the rest of the UK’s, and many of its inhabitants are isolationists. Don’t know much about the geography os the place, though; it’s not as far North as the Shetlands or Orkneys, which sound more like what your looking for.
The Shetlands, Orkenys and Outer Hebrides are definitely big on atmosphere and extreme weather. The Outer Hebrides are among the northernmost inhabited places in the world. If you’ve seen Castaway 2000, then you’ll have a good idea of what Shetland’s like.
(In case you haven’t seen it, Castaway 2000 was a show on the BBC, also shown in the US on some station or other, where 30 ordinary people, including families with young children, went to live on an island for a year. They built their own houses, tilled their own fields, set up a school, made a real community. A fascinating progect that the BBC didn’t make near enough use of).
The Channel Islands have been a part of Britain for several hundred years, back to when this country used to rule over large parts of France. The Channel Islands are the only territory we’ve kept from that time. The islands were occupied by German forces for most of WW2. They have a feudal, undemocratic system of government, are a tax haven, and have about one millionairre to every three inhabitants. Geographically, the Channel Islands are quite warm, with lovely beaches and fairly flat land.
As a first place to visit, I’d recommend the Shetland Isles, though which islands in particular I’m not sure. Most of them are only accessable by boat, and the ferries don’t travel that often, but they are still easier to get to than the Orkneys or Outer Hebrides, and have better facilities for tourists. If you’re up there in summer, hire a boat and tour the islands. (Don’t try it in winter unless you’re a highly-skilled yachtsman with a good life insurance policy).
If you like the weird and off the beaten track, try rural Wales too. I recommended Wales as a destination for a couple of American friends of mine, and they just left this week, saying that they ‘had a blast.’