I don't think the budget is a problem at all... even with Paris and London.

There is a big gray area between 4* hotels and backpacking. Consider an apartment rental for some of the more expensive places. That will allow for some meals in, to make up for the more expensive restaurants.

And don't cut out flying as an option. Intra Europe flights can be pretty cheap.

If it were my 3 week trip...

Week 1 - London (2 days) Paris en evirons (5 days)
Week 2 - Spain
Week 3 - Northern Italy