I used to live in Somerset for my first 20 years, which is a bit south of Bristol, and a good point on the way to and from Land's End. You can see that I am somewhat biassed in my route recommendations, but I have ridden round all the areas I recommend. Now I live in London, meaning I appreciate the nice roads of Somerset much more. I also did LeJoG (Land's end to John o' Groats) a few years ago.
Personally I wouldn't go as far as Cornwall unless you have a burning desire to get to Land's End as it's all horribly touristy in my opinion, though there are of course nice bits. Also I don't know how long you planned to spend on the road - the route seems ambitious if you want to spend less than a couple of months on the road. Better to spend some time in Dorset / Devon / Somerset in my view. Some ideas for you:
1. Avoid London (and getting too close to it) like the plague. Certainly don't go inside the M25. There are plenty of smaller roads slightly inland of the coast which will take you on a nice trip from Dover westwards. Main roads in Kent are very busy, so avoid long rides on these if possible. That said, small roads in Kent can be extremely picturesque. It may be a surprise, but Kent can be very hilly with regular steep uphills on the minor roads. The main road out of Dover is particularly nasty.
2. I like Brighton on a sunny day - typical English seaside resort and basically playground for London weekenders and stag / hen parties at weekends. Lots of cheap hotels and campsites, e.g. Abbey hotel, which is very basic, clean but cheap. I stayed there on a Stag weekend recently. Not sure whether they can store a tandem for you though as they are a backpacker's place really.
Alternatively you could go a bit inland for nicer country roads / villages. I don't really know it well as I don't quite get that far on a bike from home. I suspect hugging the coast would be hillier than if you go inland 10 or 20 miles. Look out for Oast houses, used for drying hopps. Now most are converted into houses.
3. Salisbury - Nice cathedral town, and worth an overnight stay.
4. South Devon / Dorset - Lyme Regis is a good places to visit for a bit of seaside. Lulworth Cove, Durdle Door are interesting from a geological point of view.
5. Don't bother with Cornwall - busy with lots of holidaymakers and caravans. Everywhere tourist / camping oriented, which is both good and bad depending on your view. I did ride through it doing LeJoG, but on the main roads, which I wouldn't do again as it's much busier now. Land's End is also worth avoiding. It's a signpost with a view, similar to many other coastal views. However it also has a car part, taccy souvenir stands etc...
6. Instead head North, round the western edge of Exmoor - Ilfracombe / Barnstaple are great places to relax and see the sea. Roads to get into Exmoor are suitably testing as it's always uphill or downhill.
7. Head East towards Exeford on the way towards Taunton. Excellent scenery. Somewhat hilly. Traffic thins as it's in the wilds, though gets busy again at Taunton. The coast road (Porlock/ Minehead) is very pretty, though can be busy at holiday times. Definitely avoid other main roads as traffic can be very busy on main roads during weekends and holidays as anywhere west of Taunton is a caravan destination.
8. Glastonbury - Interesting Tor (tower on top of an isolated hill) and all sorts of medeival / ancient legends surround the place, plus a ruined abbey to visit. Stay at my mum's B&B in Street. Obviously I am biassed, but food is better than most restaurants and she welcomes bikes. See www.oldorchardhouse.co.uk
for pictures. The levels (flat area c.2-3m below sea level) round Glastonbury / Street make good cycling country as long as it's not too windy.
9. Take one of my favourite routes out on the minor roads from Street across the levels, up Cheddar Gorge from zero to 280m elevation in 3km, riding up the route of a collapsed limestone cave and up onto the top of Mendip hills, then on to Bath. I don't like Bristol that much, so I'd recommend you don't bother, and spend the time in Bath instead as there are lots of Roman / Georgian things to see. If you must visit Bristol, I like the SS Great Eastern, a preserved ship in the harbour.
Wales - some kids threw stones at me in Merthyr Tydfill last time I was there. Rains a lot.
10. If you do decide to come back from Bath, you could stop at Longleat, visit the house and gardens, though not the safari park on a tandem.
11. You could ride across Salisbury plain to see Stonehenge, and Avesbury, which both have neolithic stone circles. Do not even think about riding one meter on the A303 though as it's basically a motorway.
11. From there you should basically follow a straight line south-east, to get you well south of Reading and avoid Guildford, other similar commuter belt towns as these are all extensions of London and my vision of hell on a bike. Route back from there to Dover shouldn't be too bad.