The British Open is the pinnacle of UK contests, a good choice. Even the band that comes 18th will have given a quality performance. The European contest is probably the only one I would recommend ahead of it.
Edinburgh is a lovely city, though as noted it isn't near the distillery heartlands, which
tend to be further North or West, with the greatest concentrations in the islands west of Kintyre and Speyside, near Inverness. Glenkinchie is a nice little distillery - we took the brass band there when we visited Edinburgh as a group 4 years ago. The centre is beautiful, centred around the iconic Castle mound. Wandering around that area will repay you - you can go to the castle (though it's a bit pricy), visit St Giles' cathedral, see attractions (e.g.
https://www.realmarykingsclose.com/,
http://www.nms.ac.uk/national-museum-of-scotland/), even see the building where we got married (the big room in Lothian Chambers, George IV Bridge)... And just enjoy the curious 3-dimensional layout of the streets (the split-level Victoria Street is well worth a look, with the pubs of the Grassmarket at the bottom). Walk up Arthur's Seat, head South into the Borders to see some beautiful scenery within an hour's drive, head East along the Forth to pretty seaside villages and towns like Gullane, Aberlady, and further along, Dunbar and North Berwick. Drop in on a brass band rehearsal - I recommend Dalkeith & Monktonhall band, where Diane (Mrs Dave) played for many years, just out of town, rehearsing in Prestonpans (I think they rehearse Mon/Thu) - drop me a line if you want a contact.
En route to Birmingham, the Lake District in Cumbria is beautifully scenic - if you are travelling by car? Snowdonia in North Wales is off your route to one side by a fair way, but also lovely. There's the Peak District in Derbyshire, closer by, which offers scenic walking.
Birmingham itself doesn't have a particularly strong reputation for touristy stuff. I've never really tested it out, so I can't comment whether that's deserved or not. The hall for the contest is absolutely first-rate, though.
Heading down the M40 to London, Oxford is not far off your route, about halfway along. The historic centre, full of colleges and their buildings, is well worth a look (though parking is expensive), and you'd be very welcome to drop in on us at nearby Kidlington band (definitely Mon/Thu) if you wanted.
