If you really want to experience the UK, you avoid London or dont do London until the very end. Same with Edinburgh really-for such a small provincial town, it has a sense of itself far in excess of its reality, and prices to match.
The true British experience is the try some of the Northern cities, then venture to the countryside which is where you see the true history and beauty of Britain. Those cities unlike London are recognisably part of the country they sit it, and are within easy reach of rural areas-Manchester is often a good place to start, especially as you can fly there from many places in the US. From there you are within easy reach of Liverpool, much of Wales, Glasgow and the Scottish borders, Newcastle, Birmingham etc, plus the vast majority of the national parks, the coasts etc.
One thing that cannot be overstated is the public transport, especially the trains. In comparison to the US, the UK public transport network has the benefit of, well, existing. Its not perfect and we Brits like to complain about the trains, buses etc, but you can get to pretty much anywhere, from anywhere else, pretty reliably, in reasonably comfort (Well, as reasonable as a bus or train can ever be) and in a relatively short space of time.
If you must go to London, its useful as a point to fly home from, and if you go there last it wont be your only or even main experience of the UK.
And frankly, although London can be expensive, it aint 2 grand for 3 days expensive-ABD is frankly not worth it looking at what they are offering (apart from anything, half the stuff they are including in this excursion, can be experienced for free if you plan it yourself, or just isnt worth doing-the State Rooms of Buckingham Palace? Dont bother, its a few portraits and a nice carpet: you wont be seeing the Queen no matter what you think, so save your time). Westminster Abbey? Paying to get in? Half the time you can get in for free if you turn up just before, after or during a service (it is a working church afterall-just keep quiet and hang around after its finished)