It's also possible that you could move house, but keep the mortgage. Might be worth enquiring. Bear in mind though that they will almost certainly try and sell you something (even a new mortgage)
You'd probably have to pay a small admin fee, and they'd need to survey the new property to make sure it's worth enough & not about to fall down or be bulldozed, but you two are already underwritten, so that's a whole bunch of admin that you shouldn't have to pay for. Sod's law you'll have to speak to a few 'computer says no' or 'not our policy, have a new mortgage' morons before you actually find out whether you can or not.
As for savings, have a look at ISAs. tax free growth, and if you are higher rate taxpayers, that's not to be sniffed at.