Here's my take, but there'll be no comments on Brighton and
Liverpool as I didn't go there. It's doable in 2 weeks, but I'd
allocate an extra day in Scotland.
London - The British Museum has free admission
(i.e. make a donation!) and can be easily done in one morning, just
be prepared to squeeze to catch a glimpse of the Rosetta Stone. Be
prepared to spend a long time queuing for the Tower, give it half a
day. I didn't go to the War Museum, no comments there.
You can split your London itinerary in 2 if you're arriving and
departing there. There is really a ton of stuff to do and see
there, and it depends on which days of the week you're there. Go to
at least one market if you can, and since the National Gallery is
free too, take advantage of it. Brighton is pretty close by and
easy to get to so you can stick a day trip in either London
leg.
Manchester - On top of the stadium tours you
can visit the Museum of Science & Industry and People's History
Museum. There are 3 free shuttle bus routes around the city during
office hours. It's a strange jumble of glass boxes and Victorian
buildings to walk around. I'd give it 1.5 days but I had only one,
so I skipped the football.
Glasgow - Plenty more Victorian buildings,
fewer glass boxes. Hard to say if you'll find something to interest
you but I went to Kelvingrove art gallery, visited the CR
Mackintosh buildings (including the School of Art and tearooms), St
Mungo's, Winter Gardens and took a walk along the Clyde in 2 days.
The Old Firm rivalry between Celtic and Rangers is also very
intriguing. Or you can use it to rest for the drive to Skye.
Skye - I'd stick it between the 2 Scottish
cities. Plenty of beautiful scenery like the Five Sisters, a great
whisky distillery in Talisker and Eilean Donan castle on the way
in. Weather may restrict what you can do outdoors however and make
things look less picturesque. 2 days will be enough, 3 will be
quite relaxed. (If you're hungry between Glasgow and Skye, Oban has
great seafood).
Edinburgh - I'd give it 2 days - there's
Holyrood Palace and Edinburgh Castle at either end of the Royal
Mile (and its various attractions like the camera obscura, St
Giles, John Knox's house and the various closes and ghost tours).
It'll take a full day, and then you might to watch the sunrise from
Arthur's Seat, or make a daytrip to Leith or St Andrews.
And since you're there, don't forget to try stuff you won't get
in S'pore like haggis, black pudding, steak and kidney pies, a real
full English breakfast and tallow-fried fish and chips (from The
Tailend in Edinburgh/St Andrews) .