Nowt wrong with getting left over there for a car guy. Interesting cars are much cheaper over there .
Some random brain dumping:
You're at Beaulieu already, so I'm sure you'll take in the museum there. I went there, gosh, over ten years ago and really liked it.
South Coast over to Cornwall is nice, Wales, is, well, nice for Wales and has great driving roads but with a PITA police presence. When I was living over there, most of my motorcycling friends gave North Wales a big berth because of the rather humourless policing out there and I don't think it has got any better. I can check with my friends in the UK.
You're at the Southern end of the country, and Scotland is a loooooong way away. I know it doesn't look like that on the map, but you'll lose at least a day each driving up there and back. If you only have a week, that would be too much for my liking unless you do something like flying from Gatwick to Edinburgh.
Car stuff, well, there's this: http://www.heritage-motor-centre.co.uk/
If you don't want to do the big city stuff, I'd be tempted to do a roundtrip taking in Cornwall, Wales, stop at the Heritage Motor Centre and maybe if time allows, make it up to the Lake District and then back south. That should take in most of the beautiful parts of the country on the Western side without going up to Scotland and is just about doable in a week. For a more relaxed trip, I'd pick either Cornwall or Wales. If you do head for Cornwall, make the detour to Bath. Great old place.
Personally I like East Anglia and went up there whenever I needed a break, usually staying at this place. However that's more for relaxation and not if you want to go sightseeing.
Another option would be to just stay South of London and do a tour starting in Kent (Canterbury is probably the best place to start) and just amble westwards. While crowded, I really like Southern England and Cornwall once you get out of the London commuter belt which these days kinda extends quite far. Either way it's beautiful and once you get past Southend it also gets a bit less crowded.
When planning the trip, keep in mind that while the UK is pretty small, it's pretty crowded and driving places takes a long time even if you can take the motorways (Cornwall, for example, doesn't have any motorways, the motorway ends in Exeter). From my old home in Ashford, Kent, about 30 minutes and 20 miles west of Canterbury, I could just about do either Manchester/Bolton or Swansea in Wales as a day trip if I left home at 6am-ish and didn't mind getting back well after 9pm.
Edit: Forgot to mention, if you have any specific questions or need specific recommendations you don't want to put in this thread, feel free to PM me.
Edit 2: Average speed cameras have become much more prevalent over there since I left. They used to use them a lot in roadworks, now they're all over the place. In fact, sometimes I get the impression the whole country is one big speed camera, so be careful out there.