Staying at Premier Inn across the river from Parliament and planning for three days in London

You should definitely buy tickets to the War Rooms in advance if you want to go there. My reaction was that it's a skippable sight (and an expensive one) if it's just the bunkers you're interested in. The Churchill Museum is very good; I spent hours there. It's definitely worth the time and money if you are interested in that part of the sight. A full visit can take 4 hours or more.

I have never been inside St. Paul's or Westminster Abbey, so I have no idea how long those visits would take or what the ticket situation is.

Edited to add: Westminster Abbey and the Churchill War Rooms are very close together. St. Paul's is about 2 miles from the Abbey and 1.4 miles from the Premier Inn County Hall. So St. Paul's is the outlier. All of those places are walkable if you like to walk. Allow extra time to get to the War Rooms, which are set well off the road and not as well-signed as they could be.

There are trains to Canterbury from St Pancras, Charing Cross and Victoria Stations. Many, many trains. Travel times vary, with St. Pancras having the fastest trains, but there is variation even from that station. You can easily check schedules at nationalrail.co.uk. There are a couple of stations in Canterbury, Canterbury East and Canterbury West. You can take a look at Google Maps to see where they are vis-à-vis the historic center; there's not much difference in the walking time to the cathedral. As you can see, CW should be called "Canterbury North" and CE should be called "Canterbury South".

I day-tripped to Canterbury by train a couple of years ago. It was easy to do. The tourist office runs walking tours in the historic center; I believe they were offered at least twice a day. There may have been more than one itinerary. The center is touristy but very picturesque; I spent considerable extra time walking all the streets of that area, not all of them being tourist-clogged.

There's a stained glass museum upstairs in the cathedral (no elevator) that I found very interesting. There's a modest fee for it. It was great to be able to see the glass at eye level.

Yes all 3 are doable in a day but you’d be slammed and exhausted dealing with the crowds and not enjoying each as you’d be rushing off to the other. Savor your few days in London. Make the time to truly experience the sites. Vergers tour of Westminster Abbey. You’ve said you’ve been to London before then you know it can be hectic.
As the Abbey and War Rooms are within easy walking distance of one another I do both in one day. See which opens first, go there, have lunch
( nearby 3 Chairman pub) or in the Abbey cafe.
Suspect you’d be done by mid day, then walk into and around St James Park or stroll along the Thames Path or hop on the tube and...

From Premiere Inn locale it is an easy stroll to Westminster Bridge then over the Thames into Parliament Square.

You are traveling in the height of tourist season so yes Book both the Abbey and Churchill War Room tickets in advance. Do them in the morning then with your Oyster card travel elsewhere for lunch and exploration.

Maybe Coal Drop Yards behind St Pancaras, or the Hampstead village then stroll in the Heath, or tube to Putney Bridge, visit Hurlingham Books, then walk along the Thames path to Hammersmith and enjoy a pint at The Dove.

St Paul’s should be booked in advance as well. Do it on another day. See it in the morning then walk to the glass covered Leadenhall Market. From there walk up to and explore Spitafields Market.

Also look at Timeout London online to see if there’s a special exhibit, event etc that might be of interest to you.

County Hall is an excellent location for walking/strolling. If you choose to amble beyond the three sites you have mentioned, you could take in the Globe, Borough Market, Tate Modern, even the Tower. You can stroll in the evening along the Thames South Bank and find some quick places to dine. Waterloo Station is quite close by and has some grocery store opportunities and bank machines. Very handy.

The Imperial War Museum is also walkable.

I don't see how one could visit both Westminster Abbey and the War Rooms in one morning if interested in the Churchill Museum component of the war rooms.

Can I just ask if you’re at the Premier Inn County Hall or the Premier Inn London Waterloo? It shouldn’t make much difference since they’re only a block apart.

Those two hotels are close! I stayed at the Waterloo Premier Inn and enjoyed that location. Easy transport connections and lovely walking areas. I encourage a verger tour at Westminster Abbey.

We are staying at the Premier Inn, Waterloo.

We just did the Churchill War Rooms and Westminster Abbey in one day, and regretted it.... Both are super interesting and well worth seeing, but both are VERY crowded. I could only take in so many crowded areas and as a result couldn't really focus on things that I would have liked to in both places. Both places herd you along with an audio guide, and it was just too much. Sadly we didn't make it to St. Paul's. Have a great trip! I'm already ready to go back and explore more!

I think the War Rooms are a can't miss attraction. The story of how it was constructed and how the workers lived is very interesting This was survival and they were quite ingenious. Then at the end of the war they just walked away. We have seen it twice. Bon Chance