Hi Simmy,
I think you've already had some good advice. When I cycled to John o'Groats from Bath, I found most of the route in the south of England boring, busy and unpleasant - it got much better once I got north of Manchester, though. Having said that, I do see the attraction of leaving your house and getting to the Algarve by pedal-power alone - I've certainly fallen to that temptation myself a couple of times! Whatever you decide, the most important thing is that you enjoy the experience - if enjoying the experience means taking the train to avoid the nasty bits, then so be it. Personally, I don't like to think too much about how many miles I do in a day; I try to think more about the scenery I've seen and the people I've met.
As for your bike, well my advice would be to get something like
this: I've had one for three years now and I absolutely love it. You will need to buy more stuff, though: front panier racks, paniers, a better saddle and a kickstand for starters.
Finally, have you heard of
warmshowers? If not, take a look at it - it's a very good way of keeping the cost down (bicycle trips always cost more than you planned!). I'm on it myself (under the name of JimP) and have thoroughly enjoyed hosting cyclists and being hosted myself. Bear that in mind if you decide to got via Amiens!
Good luck!