Two major sites- Possible in a day?

You actually mean Westminster Abbey, right? Yes, I've done this. Get to the Tower when they open (if that works for your group), have lunch there then take the Thames Clipper up to Westminster Pier for Westminster Abbey.

It will be a big day with lots to see and for your brain to process.

Certainly, doing a basic self tour of the tower, seeing the jewels, and a bit of wandering around can be done from opening until Lunch. There are probably multiple things you can add on to your tour there, but very possible to see most everything in a morning. Have lunch on your way to Westminster, then tour there. Really, unless you try to add on every option to Westminster, you are talking a couple hours max.

If you don't want to rush the Tower, which takes at least 3 hours, maybe 4 or 5 if you are thorough, you could attend evening services at Westminster, which gets you in at no charge. No sightseeing is allowed, however you experience the religious aspect of the building.

Sure but you’ll need to double check opening time for Westminster Abbey to be certain its not booked for some event.

Start at the Tower first ( go at opening time and head straight to the Crown Jewels)…when finished seeing the Tower of London walk across the road to the Tower Hill tube station and take a Circle ( yellow ) or District ( green ) train to Westminster Station. Exit. Walk over to the Abbey. Get your admission ticket.

Have lunch in between.

The Two Chairman pub is a short 5 -10 minute stroll from Westminster Abbey. Even closer is the Red Lion pub which for years was where I’d go for my first pint. Both spots would do for lunch.

I recall spending about 4 hours at the Tower on my first visit there (subsequent 2 visits have been focused on observing the ravens) and seeing all exhibits--crown jewels, touring the buildings, and doing the guided Yeoman Warder tour--don't miss that, the Warders are delightfully amusing & informative. Westminster Abbey took perhaps not more than 2 hours. There's a really good cafeteria-style restaurant at the Tower, perhaps have lunch there and then on to Westminster--be sure to book a timed ticket.

We planned our visit to Westminster Abbey to coincide with the Evensong service. When we finished our visit we simply told the docent we were staying for Evensong. We didn’t have to leave and we’re first in line when the abbey started letting people in. We sat in the quire beside the choir during the service.

It probably sounds pedantic but it really is important to use the correct and full names for things in England - so many, all over the country, have similar names, from roads to towns, to buildings, to stations.

I saw in a previous post you asked for information about Westminster Cathedral, and were told that that is the Catholic Cathedral which gets many fewer tourists.

If like here you just ask for Westminster, depending on who you ask (perhaps a person on the street, "Which way to Westminster") and where they are at the time, they may well point you to Westminster Underground station, or Westminster Cathedral, or the Houses of Parliament (often shortened here to Westminster), or Westminster Bridge, or the entire City of Westminster (yes, what looks like a Borough is in fact a city, since 1539), when what you meant was Westminster Abbey.

From the fact that you think it will take a long time I can probably deduce that you mean either the Houses of Parliament or Westminster Abbey. Either way, that is a big ask on one day. The Tower of London can easily take half a day at a trot, then you have to get all the way from Tower Hamlets all the way to Westminster (the city) which in London traffic can take an age, or the tube, and you will need lunch. If Parliament is sitting and you want a tour , or to sit in the Strangers' Gallery, you need to plan in advance - you aren't allowed to take anything in with you and there will be a strict security check, and it is either queue or have advance tickets. That's most of the afternoon gone. If on the other hand you meant Westminster Abbey, if you don't want very long queues you need to have a timed ticket in advance and plan on a couple of hours or so inside to get your money's worth.

Either way it is a very long day. I'm a member at the Tower of London and even on flying visits rarely stay less than a couple of hours and I know the place well. And it is tiring. No public elevators in the White Tower you know, I was built in 1066 and the walls are very thick.