London Honeymoon

Why don't you see if you can stretch your trip a day or two. Spend 1/2 the time in London area and take the Eurostar down to Paris for the other days. Then fly home (open jaw) directly from Paris.
Your honeymoon would be much more romantic if you had sufficient time in Paris.

From London by train, Paris is about 2 hours each way, and Manchester or Liverpool are 3 hours each way. That's a bit far for a day trip, but doable if:

1) You are very organized and have done a lot of research beforehand, about what you want to see, how to get between points of interest, etc.
2) You accept that you can only see a few things.
3) You don't mind the time being taken out of London to see other places.
4) You buy train tickets ahead of time (unless you're loaded). All of the trains for these trips are affordable if bought in advance as non-refundable and non-exchangeable. However, the prices skyrocket if bought at the last minute, and/or if you need flexible tickets that can be changed.

There are many places closer to London that are easier to see as daytrips, that don't need advance planning to such a degree, and that will be cheaper. Some examples are Oxford, Cambridge, Salisbury, Stonehenge, Bath, and Brighton; there's plenty more. However, if Paris or Manchester or Liverpool calls to you more than these other places - go for it!

I say this as someone who spent less than 24 hours in Ankara, Turkey, but had a great visit. We flew in one night, did a "surgical strike" to see a few things that interested us, and flew out the next afternoon. It worked very well, and remains a very memorable travel day.

If you want some ideas about what to see in Manchester or Liverpool, here's my trip report from September 2016: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016

I was cross-posting with David. His idea is a good one if you have more than one days of sights in Paris.

I also forgot to respond to this:

"I am fine with renting a car for day trips or taking public transit."

For the three cities you mentioned, train is the only way to see them in a day. Driving to Manchester or Liverpool takes almost twice as long as trains, and driving to Paris takes far longer (and the steering wheel will be on the wrong side). Not to mention, a car in all four cities (London, Liverpool, Manchester, and Paris) is not only unnecessary, but a huge and expensive headache.

For the other daytrips in England that I listed, a car is not necessary, is a huge hassle, and again, by the time you add in the time it takes to pick up and return a car, it won't save time. If you really want a driving vacation to visit small towns, get the car when you leave London and/or drop the car before you get to London - or you'll be sorry.

It looks like you have 8 nights total so since it's your honeymoon, I think you should give at least 2 of those nights to Paris. I agree, it's the most romantic city in the world! Just imagine having a picnic on Champs du Mars while watching the twinkling lights of the Eiffel Tower. So what I suggest is 2 nights in London to get over jet lag, then take the Eurostar to Paris for 2-3 nights, then back to London for the remainder of the nights. That will still give you time to do at least a daytrip out if London. If you're Harry Potter fans, I recommend the studio tour. You can book roundtrip tickets on the Eurostar up to 4 months ahead so you have plenty of time to do that. Enjoy the planning!

Thank you, everyone, for the replies! Unfortunately, we are not able to extend our holiday so we're kind of stuck with the current schedule. I don't have any great desire to rent a car, so it's good that won't be necessary!
My fiancé is the one interested in Manchester, mostly due to Man City and Oasis (not sure there is anything Oasis-related to see!). I'm thinking we will try to do 2 nights in Paris and then maybe Liverpool as a day trip. Thanks again!