To offer a slightly different take, if you are planning on visiting for 3 - 4 months, I'd argue that August is a great time to start as you will be visiting at the beginning of the autumn. The points that WP and Richard make about it being the busiest month are valid, so bare in mind that accommodation will be in short supply and most likely expensive (£80 pppn as a minimum if not camping). Also as mentioned, account for a fair amount of driving!
An idea for an itinerary would be something like:
First week: Hit up some touristy stuff (London, Oxford)
Mid - August: Pembrokeshire - a complete change from cities, like Cornwall but quieter (I'd argue more beautiful). Choughs, Seals and other marine life, autumn passage just beginning with common migrants beginning to filter through. Cardiff, Brecon Beacons are good tourist options too.
Late August: West Country - Slimbridge, Somerset levels. Good birding at all times of the year, friendly folk and sample some cider. Bristol, Mendips, Quantocks, Exmoor for tourist spots.
Early September: Head down to Cornwall, Devon en-route. Seabird passage picking up, and summer holidays have finished. Seabird passage west Cornwall can be incredible (Shearwaters), again passage just beginning, so common migrants beginning to filter through (pipits, wagtails, wheatear, hirrundines).
Mid September onwards: The east coast is arguably the place to be, so a few days taking in somewhere in the midlands before heading over to Norfolk would be a fair option - perhaps a stop in Cambridge to break up the journey. Norfolk and Suffolk are a must (Titchwell, Holkham, Cley, Minsmere) - migration should be in full swing now (waders, passerines, first wildfowl).
Early October: Yorkshire always a good bet at this time of year, with Spurn or Flamborough offering the potential for really impressive migration.
Mid October: Head up the east coast taking in Northumberland, another beautiful part of the country with exceptional birding. You could easily take two weeks exploring the area.
November: Scotland - so many places with a lot of history, and great birding!
As an alternative, if you could change your dates, March - May would be perhaps as good, as the birds are easier, with the change between wintering and summering species more apparent.
Hope this help!