Thoughts on 2 1/2 days in London

I think your plan is realistic and do-able in two and a half days. The time to do each one is as follows, but is only an estimate. (You may want to linger longer than I did at some of these sites.) The Churchill War Rooms may take 2 hours. St. Paul's Cathedral may take 2 to 3 hours. British Library may take 2 to 3 hours. Your other destinations (the National Gallery, Tate Britain, British Museum) will each take much longer. The British Museum will take the longest amount of time, because, as you know, it's huge. You could literally spend several days there. You should visit their website before your visit and determine what objects and rooms you want to see the most, so that you can hit those rooms first. The British Museum is open late on Friday night until 20.30.

Here is your guide to what is on display and in which rooms:
http://www.britishmuseum.org/explore/galleries.aspx

One of the highlights of the British Museum, the Rosetta Stone, is in room 4. You will want to see that, as well as the numerous Greek and Egyptian artifacts (many rooms). Roman Britain artifacts, room 49. Sutton Hoo, room 41, another of the great treasures of the British Museum.

You shouldnt feel too rushed with this plan of attack ...
Full Day
Churchill War Rooms => St Pauls (perhaps for 1230pm service) => British Library (stays open til late)
Full Day
Tate Britain => British Museum
Half Day
National Gallery

I would highly recommend Tate Modern, even for a quick look, while you are at St Pauls.

I'm going on a Best of England in Sept too, so I have been working on what sights to see on my extra days. On the last day of sightseeing we will have a half a day after we do the Tower in the AM. I am going to work in St Paul's that afternoon before the farewell dinner.

Of the remainder from your list, I personally would prioritize them as #1 Churchill War rooms, #2 British Museum, #3 National Gallery, #4 British Library and #5 Tate Britain. That is just how I would rank them for myself. Your tastes and interests may be vastly different than mine! If you have 2 full days left, you could do the Churchill War Rooms and the National Gallery the same day and the British Museum and British Library the same day, with the Tate Britain on the extra 1/2 day (unless you are counting your last tour day as your remaining 1/2 a day?). I would move the British Library UP the list if you are going to be there before their current special exhibition of the Magna Carta closes, but I think that the last day on that is Sept 1...and yes, if I had done enough research ~before~ I bought my plane tickets, I would have flown early to see it!

I did not look to see if any of your choices are open late on Thursday night, if indeed you are doing a Sunday tour departure. Some of them depart on Tuesday so you would be winding up in London on a Saturday afternoon.

I'm so looking forward to this tour!

FWIT, I agree with Emma, particularly her 2nd sentence.
Sometimes in Europe travel, doing less in a day increases the quality of your experience.

Ah, Pam! But you'll get there in time to see a great exhibit opening at the British Museum on September 24th: "The Celts; Art and Identity". The reason I am going there this fall.
http://www.britishmuseum.org/whats_on/exhibitions/celts.aspx

Rebecca!!! I had not seen that exhibition. OK, now I need to finalize my plans and order my ticket to that. So, looks like there are tickets for the 1st entry time at 10A on either day I have open, lol. Thanks so much!

Pam, you're welcome!

Unfortunately all the places mentioned are quite scattered across London. However, as they are all quite 'dry' experiences (museums and galleries) you may be thankful for some time out with the travelling in between.
The British Library and British Museum on the same day would be a sensory overload for me.
War Rooms to St Pauls by District Line (Westminster to Blackfriars/Mansion House)
St Pauls to Britsh Library by Northern Line (Bank to Euston)

Thank you for all the helpful suggestions! You've made my planning much easier.