Andermatt for the Swiss Alps is a good recommendation, but you can ride all the obvious circular routes fromt here in a couple of days. Or head for Grenoble in France and pick roads from there. I love the roads and scenery in the Tyrol and the Dolomites, but they're that bit further away and in four days it's not really practical.
It takes a day to get there and a day to get back (and that's from Calais/Channel). Add 3.5 hours to Notts/Derbyshire from there... If your four days include getting there and back, then you're going to be knackered - probably too knackered to really enjoy it. But assuming your four days off work are next to a weekend, then you'll have six days altogether and it might be more fun.
Riding for ten hours/600 miles in a day is fine, but in my experience riding ten hours/600 miles two days running is a whole new level of pain.
If you really are limited to four days, my honest advice would be to forget the Alps. Head for the Black Forest and/or Vosges, which are more easily reached. The roads are almost as good - some would say better because there are no stretches of 36 caravan-strewn hairpins to negotiate, which aren't everybody's cup of tea - and the lower altitude means you won't have the same risks of snow, hail, run-off and so on.
Channel crossing - I've only ever used the short crossings to France. Tunnel is quick and easy, ferry gives you chance to have a sit down and some grub. Bikes are ususally first on/first off the ferries, always last on last off the tunnel but the tunnel is still way quicker and simpler - no tying down etc.
Routes - time may well dictate that you have little choice but to take motorways all the way. You can avoid tolls if you go down via Belgium, Luxembourg and Germany. Or use the French autoroutes if tolls don't matter too much. Decide how far you think you want to ride on the first and last days, pick a town nearby, and book a hotel nr the motorway. I use a chain hotels on France for reliability, ease of booking and so on - B&B being most used. They are a little soul-less, but clean, cheap and conveninently located.
Finally, look at
ALPINE ROADS for detaisl fo what passes are wehre, but remember that any Alpine road is going to be a thousand times better than any road here, so just because it is an "average" alpine road doesn't mean it's no good.