Harewood House / Castle Howard - public transportation logistics?

It's a three hour drive for the first part if there aren't any issues - but through a part of the country liable to have issues with congestion, roadworks and more. You could get lucky or it could take longer.

You would be looking at a train to Leeds then a couple of buses to Harewood House (probably 3h30m) - then a bus back to Leeds then two buses out to Castle Howard (4 hours), then another buses and train combo via York (4 hours plus)

The drives would be 3 hours plus (Birmingham to Harewood), 1 hour (Harewood to Howard), 3 hours plus (Howard to Birmingham)

Even with the risk of some extra time due to traffic, I would take the car option - especially as you say you are comfortable with that.

Another option though would be to use train/bus and after Howard go to York and get a train back to London

Is there a reason why you don't want to change lodging to someplace nearer to both Harewood House and Castle Howard, such as York? It seems you could stay in (for ex.) York on Sun-Mon-Tues nights and then go to Heathrow on Wednesday without having to backtrack to Birmingham. Or maybe I misunderstood what you meant by "position" for the Monday visit.

In any case, I do agree with the previous poster that renting a car may be your best option. If you rented out of Birmingham and returned the car at Heathrow, I don't think there should be a significant extra charge.

Thank you for the responses so far and to clarify.

I would stay in Birmingham overnight only on Sat 02 Sep, and would plan on staying Sun/Mon nights somewhere local to
Harewood House or Castle Howard or in between the two. Any suggestions for a place to stay (near/in York?) Sun / Mon
night would also be appreciated.

I would drive from Birmingham north on Sunday, stay overnight in the local area, see Harewood House Monday, Castle Howard on Tue, and then return the car to Birmingham to save the drop off fee and be in better positioning to get to Heathrow on Wed no later than 2-3 pm timeframe.

I thought about the train from York direct back to London and then Heathrow and partly I would prefer to avoid London transfer/hotel stay. These 3-4 days are only part of my trip -- I'll be in London for a few days (and other parts of England starting 22 Aug) before I leave for Birmingham to see a friend.

From York there is a bus every two hours direct to Castle Howard- Castle Line, run by York and District.
For Harewood use Bus 36 from Harrogate to Leeds.
As so often the complexities of public transit are being massively overstated.

But assuming you are driving why not stay in Harrogate or Knaresborough, rather than York?

My thoughts about staying near York (vs. Harrogate or Knaresborough) would allow me to be in between Harrogate and Castle Howard for two nights instead of staying one night in Harrogate and then staying closer to Castle Howard for the second night. Or staying in Harrogate for both nights and having a further drive to Castle Howard for the Tue visit.

HOWEVER, I just discovered that Harewood House will be closed for the week of 04 Sep (I was planning to visit Mon 04 Sep). I need to find a different activity for Monday 04 Sep. (I would still like to visit Castle Howard and they are open on Tues and not Mon).