Does this seem like a reasonable London itinerary or add a day?

I don't know how much time you have after London, but you are trying to pack a lot into a short time. What will you do with your luggage before checking into your hotel on Day 2 and after checking out on Day 5?

We are just doing backpacks, going as minimal as possible so we would just keep them with us.

Heads up: Big Ben within the Elizabeth Tower is under scaffolding. You won't see very much.

Day 1 isn't really a day in London....Your first full day sounds like a smart plan. Walk around when you finally arrive as that is a good way to handle jet lag.

I think Day 3 doesn't give you enough time in Westminster Abbey especially given the amount you'll spend for the entrance fee. I would allow at least 2 hours. As for Buckingham Palace, do you have tickets for an inside tour assuming you're traveling when it is open to the public? If not, there isn't much to see so maybe you can walk over there after lunch.

Have you considered the evening of Day 4 as an opportunity to see a play or musical?

On your last day, if your train departs for Paris at 4:31, make sure you arrive at St Pancras early enough to go through security and board the train. Others here with experience on the Eurostar can advise arrival time to allow for the security lines.

I'd suggest trying to squeeze in the British Museum--free attraction.
We also enjoyed Churchill War Rooms at the Imperial War Museum.
Victoria and Albert Museum is another major museum to see when you have time.

Yeah I saw that the Elizabeth Tower is being worked on so it will just be a walk by and a few pictures. I am going with two older kids, I don't think we will do a musical/play.

Buckingham Palace will be open when we go this summer, someone suggested Windsor Castle instead maybe? As for Wesminster Abbey I can add some more time there.

We will be staying in Clapham so it is about 20-30 minutes each day. I just have breakfast/lunch/dinner in there as a general idea but will just eat as we go and are hungry. I guess my concern is getting off the plane in Gatwick we are probably going to be hungry and two of us get hangry so having to make it through immigration and train ride seems like a bad idea.

I think David meant to type "...the Churchill War Rooms and the Imperial War Museum." They aren't close together.

Hah no worries, I figure once we are in town we will just take public transport to get from place to place and can rearrange stuff as needed to make more sense logistically or on different days.

So the British are saying "hangry?" C'mon! I thought that word was all ours, like "lining up" as opposed to queuing. Nobody wants to go to England to hear them speaking American English.

I long for the days when Paul McCartney's father, Jim, advised him to more properly sing, "yes, yes, yes" in She Loves you so as to not debase the language. We need more Jims. The charm is the differences.