Tour of London and Scotland

Is it possible to do Bath, Stonehenge and Windsor Castle the same day?

No.
Windsor is very near to London. It is easy to reach by local train from London. Best done as a day trip from London, either half day or full day.

Stonehenge is difficult to reach, it is in the middle of an empty plain!
You need either a car, or a bus tour, or there is a public bus from Salisbury. As Salisbury has a railway station, and is worth a few hours, the best idea is to combine Stonehenge and Salisbury in one day, and then either double back to London (day trip), or continue by train to Bath, arriving at the end of the day.

Bath cannot be combined wth Stonehenge, Bath needs at least a full day.

If you are combining southern England (London) and Scotland, be sure to include a night in York. Well worth a visit, old city, cathedral, walls, Roman bits, . . . York is half way between London and Edinburgh, and on the rail line between the two (2 hours from either), so it fits in nicely.

Given that the much of the UK is the easiest place to travel on your own, I would plan this trip myself using trains. Tour itineraries rarely include everything you want to see and invariably contain some places you're not interested in-although my experience has been that the latter is a good thing because it takes me to places I would never have gone to on my own.

We've taken Taucks 14 day England, Scotland Wales tour which starts with Edinburgh and Sterling castle. Then Hadrians wall, Chester, 2 days in Wales, Stratford on Avon, Oxford, Stonehenge, Bath, Windsor ending with 3 nights in London. Its was a fantastic tour though I would have liked a little more time in Scotland but you could do that by arriving early and doing some sights on your own. We've taken several tours and river cruises with Tauck. Pricier than RS but they really take care of you and the hotels/meals are tops. The hotels in Bath and Oxford were right in the heart of the city and the Savoy in London was amazing.

England and Scotland are both great. Doing the two in 16 days is going to mean missing out on a lot and spending a good bit of time being transported from place to place. Rural Scotland is lovely, but it takes time to see it, and it is indeed challenging if you are neither on a tour nor driving a rental car. I'd start my planning with Scotland and see how much time I had left for England, which could be a separate trip. I was frustrated with 26 days in Scotland this year, though I spent more time in Edinburgh and Glasgow than most people would. On the other hand, I can't do rural walks and have zero interest in whisky.

You might take a look at Rabbie's tours in Scotland. They seem to be a respected company. I haven't heard or read anything about their tours outside Scotland, but I see they offer some English tours as well. However, as already mentioned, England is really very easy to visit on your own.

I feel I should warn you that there is more demand than supply when it comes to summer lodgings in much of Scotland. I had difficulty finding rooms for July this year, though I was looking about 7 months in advance. You are smart to be thinking about this now.

Trafalgar has great tours, check them out.

Rabbies is a great idea. We did their one-day tour of the Highlands from Glascow and are signed up for one of their 5-day tours in England in some areas that aren't easily accessible by train. However Rabbies multi-day tours are not an option unless you're a light packer, as they use small vans and only allow a carryon and a personal item.

I have to agree with a previous poster- you can easily do this on your own! We LOVE touring the UK on our own by train (and the occasional local 'coach'/bus). With 16 days you can do a really nice trip (by the way- does that include your international flying days or is it the # of days on the ground?). Here's a rough suggestion
Fly into London-
5-7 nights with possibly 1-4 day trips (Windsor, Bath, Stratford-upon-Avon, Winchester, Canterbury, Dover, and others are all possible)
train to York- 1 to 2 nights
train to Edinburgh- 2 -3 nights
train to Aviemore (or other small town/village in the Cairngorm Mts. with distillery day trip) 1 t0 2 nights
train to Inverness 3-5 nights with 1 or 2 day trips/excursions
fly home from Inverness

My sister and her daughter did a Globus tour on that a few years ago. They had a great time.

Here’s one of their tours:
https://www.globusjourneys.com/tour/britain-uncovered/gf/?nextyear=true

Since you are active and want to see the countryside you would not be sorry if you drove. We did a 21 day thru Wales, England and Scotland and a 15 day driving trip through England and Scotland. If we can do it you can, too. You carry your stuff with you. You don't have to haul luggage thru railway stations. You can drive right up to that castle. Look online into the many admission passes available. I like the Nation Trust UK Oversea Visitor Touring Pass available online and can picked up at one of the hundreds of sites available. I happen to enjoy the greater variety available in England. Scotland is more open space; but lots of heather may be just your thing. Bon Voyage.

Thank you for all the great suggestions!