Yes, scale back. No matter if you will be driving or going by rail, that itinerary will have you spending most of your in transit. Better to focus on your priorities.
Yes, you can see something of London in one day, but not much. Especially if that day immediately follows a long flight. If you only have a mild interest in London, consider skipping it entirely and going directly from your arrival airport to Bath for your first night.
Ditto Edinburgh. If you will be satisfied with a quick trip around London or Edinburgh on a sightseeing bus, you're good to go. But, if you really want to explore either city, you'll need more time.
Oxford and Bath both legitimately merit the better part of a full day's visit,
If you are driving, your route from Bath to Oxford can go through a portion of the Cotswolds, which can easily merit a couple of days.
While you can usually make excellent time driving on the Motorways, like freeways everywhere they deliver the least connection with the country you're visiting. I.e., they're not much fun. In the Lake District and Scotland, you will typically be driving on roads that will limit your overall speed to half or less that of the Motorways.
If you are particularly interested in Scotland, consider flying from London to Edinburgh or Inverness, or rail to Edinburgh (about 4 1/2 hours). Then, by car, loop around part of the Highlands. E.g., from Edinburgh/Glasgow, drive the A82 through Fort Williams. Then west to Mallaig, ferry to Skye, back east on the A87, then north on the A82 along Loch Ness to Inverness. Or start at Inverness. Figure on 3-4 nights. Then get back to London and spend some time there before flying out.
