But otherwise, there's so much to see in those regions.
Down in Cornwall, you've got wonderful coves and seaside towns - St Ives, yes, but also places like Mousehole, Mevagissey, Padstein Padstow, St Mawes, Fowey, Looe... Then practically the whole of the Lizard Peninsula, the Minack theatre, St Michael's Mount, and then you can add in the more touristy places like the Eden Project and the Gweek Seal Sanctuary. Add in the Isles of Scilly, and I could quite happily take up a whole fortnight in the West Country.
I assume from your description that you'll be driving. Cornwall is a weird place for driving (although personally, I normally get the train down and rent a car). It's kind of cut off by the single-carriageway section of the A30 over Dartmoor, which is a huge bottlenecks on busy days (usually Saturdays, as that's changeover day for holiday rentals). Then you basically have the A30 as an artery, and travelling along it seems absurdly quick. The other trunk roads like the A390 are normal speed, but then the back roads, which you'll see plenty off, are often single-track and insanely slow.
Coming up, you pass Bath and Cheltenham, and then have the option to head into Wales - although how deep you get in depends on timing. Again, if you're near a main road corridor (like the M4), driving is fast, but once you're in the back roads, it's very slow. Getting to the (very similar to Cornwall, but pleasingly tranquil) Pembrokeshire Coast takes a while - and heading up from there towards Anglesey and North Wales takes even longer. You could head up the Wye Valley and Welsh Borders as an alternative - or take a route through the Brecon Beacons.
Further north, and you're into the big cities of the North-West. There are definite points of interest around here, but it depends what you're after. If smaller, historic towns are more of a draw, try somewhere like Chester. But then, you may be drawn to Liverpool - I don't know.
The next obvious point en route is the Lake District. It's an incredibly popular National Park - though not without reason. There are upsides (a well developed tourist industry) and downsides (very busy at peak seasons). I tend to find that accommodation inside the National Park is fusty and chintzy, so stay just outside, somewhere like Kendal (a lovely little town) or Cockermouth. Highlights for me (apart from the obvious lakes and mountains) are the smaller towns like Hawkshead, and places like the Grasmere Gingerbread shop, or the surprisingly interesting Beatrix Potter house.