Where are you coming from? That might change the order of when I visited each site.

For the Newport part of the trip I would want to do that with the maybe Blithewold as they are both in RI and do an overnight somewhere in RI. Do go to the Breakers and at least one other mansion.

Cape Cod National Seashore is almost as far as you can get on the Cape from Thornton Burgess Museum. The whale watching goes out of Provincetown, so you could plan that on the day you do your visit to the Seashore, either before or after your whale watch.

Even though the places you want to visit are not far apart in miles, it can take much longer to get between them than you may anticipate if you are used to driving in less compact/congested parts of the country.

When are you going?
Check out the dates of Waterfire in Providence (google it but I think they're doing it on 7/2 and 7/16, at the least) - fantastic experience. Providence has Benefit Street, packed with beautiful old houses and some great walking. There are organized walking tours I think through the historical society. again, google it for more info. How old/able to walk long distances is your mother, or for that matter, are you? Lots of good restaurants in Providence. Italian district (Federal Hill) has great pasta shops, etc.
IMHO, just one of those Newport mansions is enough, but each to their own. Totally over-the-top, "look how much money we can spend" palaces.
I've never been to Blithewold but have always wanted to go, the gardens look fabulous. Do while you're in RI. Where are you flying into/out of?
Cape Cod: depends when you'll be here and the weather. July and August are high season, everything is crowded, 6A will be jam-packed and very slow (I assume you have a rental car) but 6 is boring, nothing to look at. Thornton Burgess museum and grounds are cute and worth a stop on your way to the outer Cape, but if you're staying in Seeconk, you should check with your inn regarding how long of a trip it would be returning at the end of a day if you intend to just drive out and back in one day.
I haven't ever been on whale watching cruise, I get seasick. If you do decide to go, check for conditions.

If I could only choose three Newport mansions, I'd go with The Breakers, Marble House, and The Elms. Most folks find two or three mansions are about right for a day's sightseeing. Also walk the Cliff Walk, though note that the lower one-third of it is in primitive, rough-going shape and not for folks with mobility issues, baby strollers, and the like.

The Thornton Burgess Museum is set in a house he lived in. You can see the first floor only, which has some book illustrations and memorabilia related to the books. I found it all pretty underwhelming, so I'd go only if you're a big Burgess fan.

Continuing on -- two of the best tourist attractions on Cape Cod are actually not far from the Burgess House, the Sandwich Glass Museum and the Heritage Museum and Gardens.

The best Cape attractions are actually the outdoor things to do, especially beaches (notably those of the Cape Cod National Seashore) and hiking areas (the National Seashore area has several trails, as do places like Nickerson State Park).

In Providence, will definitely second having an Italian meal in the Federal Hill neighborhood. Also consider the RISD Museum of Art, but note that they close for the month of August, and the State Capitol Building.

How many mansions to see in Newport depends on how much of a house tour person you are, but in any case even 3 in one day can be a lot. They all start to blur after a while.

Assuming you're coming in the summer when all the mansions are open, I would suggest the Breakers and maybe one other smaller one for contrast. There are so many other things to do in Newport to break up the day, walk the cliff walk, drive over to Fort Adams, eat, shop.

Right now only a few of the mansions (Breakers, Rosecliff, one other I think) have audio tours that let you go through the rooms on your own and get as much or as little detail as you want. Though I usually enjoy docent led tours, I really enjoyed the audio.

A tip about pricing, there are several pass choices, The Breakers is the most expensive, but you can add additional houses for just a few dollars. One sort of secret I found was that (at least as of a few months ago)you can send a phone text message to the Newport Preservation Society and they send back a discount code that lets you buy two of the 5 house passes for the price of one. That works out so that even if you were just going to the Breakers, it would be cheaper to buy the passes! The passes don't expire, so if you come back on another trip or know someone who's comng to Newport you can use the rest another time.

I like Blithewold and it's nice, but not along the scale of any of the Newport mansions. The gardens were interesting (bamboo forest?) but I don't know if I would call them impressive. The town oF Bristol is a nice, though, and there are good places to eat, so you might want to go anyway. I would go to Tiverton on the Blithewold day. It's pretty easy to get between the two, and Newport really deserves an entire day