Tube versus Heathrow Express will always be a controversy! Most favor the Tube but some of us cannot stand it for the long ride from Heathrow, somwe buybour Heathrow Express tickets well in advance to get them for £5.50. Those are 90'days ahead so too late for that.
This page from London Toolkit explains the options:
For transport within London, the general recommendation is an Oyster card or contactless payment if you credit card is enabled for that. In my mind, the only reason tomget a Travelcard would be to use it for the 2-4-1 offers, but you need a 7-day paper Travelcard for that. Not cost-effective for a 4-day visit. Besides, you cannot get skip the line tickets with the 2-4-1 offers.
I don't know how bad the Churchill War Rooms are now, but last year it was pretty much mandatory to get advance tickets with a timed entry. I don't know if that is still true but someone else will know. We went before the hoopla (2016) and got a combined ticket with the HMS Belfast which is amazing to tour ( beware, lots of steep slippery ships' ladders).
I would definitely get Advance tickets with timed entry for Westminster Abbey. The lines are long and the interior gets very crowded. Book the earliest entry slot ( 9-11 am) and arrive early to get inside with the lowest crowd factor. We went in right at 9 and by 11 it was jammed.
I highly recommend includingnthe Jubilee Galleries with your Weatminster Abbey visit. The staircase structure is beautiful and offers interesting views out windows as you ascend. And the galleries themselves are a great vantage point from which to view the interior of the Abbey. It is an additional £5 but well worth it.