How should I go about removing these two walls?
Seeing as you're in the UK (lots of well-enforced building regulations) and you're wanting to remove at least one load-bearing wall (the old external wall), you absolutely need to bring in the pros and start dealing with all the hassles of permitted renovation. I'm not sure if you'll be allowed to do any of the work yourself, but if you are, it will undoubtedly be under the supervision of someone with very comprehensive liability insurance. :)