UK trip, late April-mid May

That's a really broad question! Ok, first off - can you do it? Absolutely. The question, of course, is "what do you want to see/do while you are there and what is your travel style?" and that isn't something anyone here can answer for you. Clearly, there is far more to see/do in England, Ireland and Scotland than anyone could cover in 3 weeks - which means you need to come up with some priorities.
So my suggestion is to pick up some guidebooks in your library and bookstore and do a bit of reading and/or watch Rick Steve's videos- highlight anything that is of interest - and then make a list!
Fair warning - the list will represent far more than you have time for- and that's ok. Once you have your "list" prioritize it. Once you've done that then you can begin to put together an itinerary - and that is where this forum REALLY shines - the collective experience here will help you evaluate if the itinerary you want will possibly work for you - in terms of if it's doable, the best way to get from A to B, how to navigate certain sites, suggestions for lodging, food, etc., etc.

Are you talking about this year? If so your itinerary may well depend on where you can find accommodation. Popular places like Isle of Skye get booked up months in advance.

Three weeks seems a long time but in fact isn't really if you are talking about visiting England, Scotland and Ireland. Taking out 1-2 days to recover from jet lag, a day travelling between England and Ireland, another from Ireland to Scotland and a third back to London begins to cut this down to just over two weeks. Don't underestimate how long it actually takes to travel beteween destinations in the UK or Ireland. Distances may not look that great, but driving/travel times will be longer than you might expect. You talk about 'most of Ireland' - this needs a week MINIMUM, and even then you will be seeing it mainly through the window.

Seriously think of pruning this down and just visiting TWO countries not three.

WAY to busy schedule.

You are traveling too much and wasting a lot of time traveling. I recommend that you do some detailed research on places of interest and how long you should stay there as well as the travel time involved.

We have been to the British Isles several times and love all the countries. Our ancestors largely came from these countries.

On our last trip to Britain, we were planning four weeks driving around with a rental car and skipping London this time.
The more that I researched, the more it was evident that we needed all four weeks just in England and South Wales.

Here is my detailed review of our trip.
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

We visited places were our ancestors came from, including a small village in SW Wales where my paternal ancestors came from in 1716. The British countryside is amazing with so much to see. I suggest that you skip Ireland and Scotland (do them on another trip).

I had driven in the UK once before and did it again, but you really have to focus on what you are doing, especially at intersections and making turns. My wife was great helping with navigation (also, we had a nav system on the vehicle) and telling me "stay on the left."

Some of our favorite places:
Bath, Stratford Upon Avon, Tenby (Wales), St. Davids (Wales), York, Durham, Winchester and Hadrian's Wall.

Some tips for driving:
For most days, try not to drive more than two hours a day.
Traffic in Britain will be heavy, especially in Southern England and especially anywhere near London.
Gas (Petrol) is expensive, consider renting a diesel car(better mileage).
Be sure and have coins for parking and toilets.
When booking lodging (with a rental car), consider where you will park. Try to find places with free parking near the downtown area, so you can walk to key sites. We stayed mainly in B&Bs and loved the great breakfasts.

Thanks for the tips! Yes, it's beginning to seem like we should cut out at least one country.
By "most of Ireland", I meant that most of our time in Ireland, we are planning to rent a car, realizing that we will only be able to visit a very few places.
We may have to cut out Scotland this time, and save that for another trip, as we would also love to explore northern England as well.

Are you planning on Northumberland? If so you could easily add a few days in the Scottish Borders to the trip.

Here’s what I would do:
Four or five nights in London. Take a direct 1h 30m train to Bath for the day.
Take a direct 2h train to York and sleep there for two nights.
Take a direct, 2h 30m train to Edinburgh and sleep there three nights. On day two, rent a car a drive to St. Andrews (1h 30m).
Take a direct, 1h train to Glasgow and sleep there two nights.
Fly nonstop to Dublin from Glasgow.
Spend three nights in Dublin and on day two take a day trip to Bru an Boinne. You can arrange a tour at tourist information office.
Rent a car and drive 2h 15m to Kilkenny. On day two tour the Rock of Cashel.
Then drive 5h to Dingle and sleep there two nights. On day two drive the peninsula.
Drive to Doolin for two nights (4h 30m) and on day day two visit the Cliffs of Moher. The next day return car at the Shannon airport (1h 15m) and fly nonstop to London.
I would cut out Liverpool / Manchester.

That is still an intensive itinerary.

Also don't bother to hire a car to drive from edinburgh to St Andrews. Catch the train to Leuchars and bus from there....