Tower of London, St Pauls, Buckingham Palace in 1 day

You’ve missed out the time it takes to walk from the Tower to St Paul’s, and you’re not allowing enough travel time to Buckingham Palace.

Additionally you have no contingency time for queueing for loos, having lunch, having a sit down etc.

I’d take St Paul’s out of the equation, or catch a bus from the Tower that goes past it so you can see the outside. You really don’t have time otherwise.

I don't think it will be possible in the time you have allotted to see St. Paul's, unless you leave the Tower earlier. If you have only this day to see this, have you considered doing an Evensong at St. Paul's? You can't walk around the building during the service, but it takes place in the most beautiful part. It was one of our favorite things on our last visit, sitting beneath the beautiful dome listening to choir music. This would allow you to see the Church after Buckingham Palace instead, if you can't squeeze it in otherwise.

How many days are you spending in London?
I think seeing all three is much too much for one day. Do NOT drop Buckingham Palace!!! First one to drop would be St. Paul’s. See it another day.
Buckingham Palace: We spent two hours there inside the Palace which was faster than most others. It was a real highlight for us after three other trips to London. We were done with the Palace way before our Garden tour started and told a guard of our dilemmas. They put us on an earlier Garden tour. It is mainly a walk through woods, wildflowers , a rose garden. Hard to believe you are in the middle of a major city. It was OK but the Palace audio tour was excellent.
And I usually do not enjoy touring palaces.
St. Paul’s may take up to an hour. Do not need to buy tickets in advance. We didn’t climb up to top but did tour the Crypt below where many famous British are entombed. I like St. Paul’s but adore Westminster Abbey and always visit it when in London. I am Episcopalian, maybe that makes a difference.
We have been to the Tower of London twice, once with grandkids along. We didn’t have to deal with crowds either time. We didn’t spend three hours either time and covered it all. I love seeing the Crown Jewels on display.

What time of year and day of the week?

Looks like Sunday and Monday are 10:00 openings, the rest 9:00 opening. If on Sunday, expect larger crowds than midweek.

Your instincts serve you well. If you went on a day ToL opens at 9:00 and the stars aligned, you might be able to make it, but it would be a mad dash.

I recommend skipping St. Paul's and catch Evensong in the evening. If you are only allotting 1 hour, you wouldn't see much more that you'd see at Evensong. I've seen it both as a ticketed visitor and at Evensong. It's really helpful if you are of the Anglican/Episcopal faith to be able to follow and participate in the service.

I understand the desire to see as much as possible, but personally I don't like to schedule more than one activity a day that demands timed entry. Then, I want to make it the first activity of the day so that I'm not under pressure to be in a certain place at a certain time.

Edited to add: The walk from the Tower of London can be great, but to really enjoy it, give yourself at least and hour. A straight walk with no dilly dallying takes 30 minutes at a moderate to brisk pace.

I can understand wanting so see everything but I have to agree with everything else that you may be better served by cutting out St Paul's. Personally, we spent about 5.5 hours at the Tower and I would have willingly spent more. The Tower does open at 9:00 on Tuesday-Saturday, not sure if you're plans can be shuffled a bit to get there on a day it opens earlier. We didn't buy our tickets in advance and got there at 8:30 on a Thursday morning and were the first in line for tickets (3rd week of September, 2019).

Thank you for all of the great feedback. We will book the Tower of London for 1st thing in the morning and Buckingham for late afternoon. I am sure we can sneak in St Pauls for an Evensong on one of the other days we are in town.

Thank you for all of the great feedback. We will book the Tower of
London for 1st thing in the morning and Buckingham for late afternoon.
I am sure we can sneak in St Pauls for an Evensong on one of the other
days we are in town.

That sounds like a plan!