Bear in mind that most decent hotels in London either have an umbrella in the wardrobe, or concierge have a stash you can borrow.

I've never stayed at the Sofitel, but I'd assume it's no different.

If it is raining, NWIFlyer's plan could be amended slightly by taking the Thames Clipper boat from the pier at the Tower of London down to Waterloo or Westminster, rather than walking it.

One other thing that often gets overlooked is the late sunsets we have in the summer. Mid-may it's 8.45pm

Also remember that British Airways strictly enforce check in times at Heathrow.