3 weeks in Great Britain

Your first issue is that staying in somewhere like Tillicoultry without a car will either mean getting into Stirling every day by bus I assume before you can start your day trips or paying someone to come and get you.
How are you getting from Tillicoultry to London? Train from Stirling is the obvious route, in which case it would make sense to stop on the way to see some of the stunning sights too far to day trip from either. Two obvious places would be the Lake District taking the west coast route or York taking the east coast route. There are of course places to recommend on either route that would easily fill your whole three weeks.

We have been to Great Britain several times and love England, Wales and Scotland.

Our best trip was a four week drive tour in S. Wales and England (we didn't go to London).

There is a lot to see in all three countries.

Some of our favorite places are York, Cambridge, Salisbury, Winchester, Bath, the Cotswolds, Stratford Upon Avon, Durham and of course London in England.

In Wales, Conwy in N. Wales and several places in S. Wales, which is loaded with castles. St. David and Tenby are worth a short visit. From Cardiff you can see the Museum of Welsh Live and some great castles.

Scotland, Edinburg, and the area north of Edinburgh, including St. Andrews. Also, Loch Ness and Inverness. The Scottish Highlands in the west are very scenic as well.

You can't do it all in three weeks. Driving on the left is not so bad, you just have to be careful and not speed. British drivers are polite and not aggressive. If you don't drive, you will miss a lot of scenic places, unless you find a great tour.

Tillicoultry to Stirling by bus is easy- every half an hour from 6am to 9pm, only takes half an hour, so that isn't a problem. And the Bus Station is right across the road from the railway station.

And Stirling has 2 through trains a day to London.

So start from a principle that Tillicoultry is a viable base.

Good day trips from Stirling by direct train: https://www.nationalrail.co.uk/live-trains/ then click on Plan a Journey at the top of the screen towards the right.
Edinburgh (1h)
Glasgow (45m)
Pitlochry (1h 15m)
You can also take a direct bus from Stirling to Queens Crescent in Aberfoyle (1h). You’ll need to call a taxi to finish the journey to Loch Lomond & the Trossachs.
Visiting the Lakes District is best by car. Instead take a direct train from Stirling to York (3h 30m) spending two nights in York. A good day trip from York is Durham by direct train (1h). Take a direct train from York to London (2h 15m).

Great day trips by direct train from London to:
Bath from London’s Paddington station (1h 30m).
Stonehenge from London’s Waterloo station to Salisbury (1h 30m) followed by a taxi to Stonehenge.
Cambridge from London’s Kings Cross station (1h 30m).
Oxford from London’s Paddington station (1h).
Chipping Campden & Moreton-in-Marsh from London’s Paddington station to Moreton-in-Marsh (1h 45m) and taxi to Chipping Campden. Afterwards visit Moreton-in-Marsh.
Warwick from London’s Marylebone station (1h 30m). Don’t see direct trains on Sun.
Coventry from London’s Euston station (1h 30m). Don’t see direct trains on Sun.
By direct bus from London’s Victoria station to:
Windsor Castle bus 702 to Windsor Town Centre, Theatre Royal (1h).

Visiting the Lake District is totally viable by public district. (it is the Lake, not Lakes, district)

You do not need a car.

In fact using a car in the area is a liability, especially in peak season, it just adds to congestion, lack of parking and parking is expensive when you can find it.

I don't know how much evidence and timetables have to be produced to show how good public transport is in the Lake District.

Of course there are direct trains to Coventry from London on a Sunday, just as there are in the week. That is a major inter city line, the same goes for Warwick. Looking on a random Sunday when there is engineering work on (with a rail replacement bus) is not giving proper advice, or real knowledge.

Coventry doesn't actually appear on these forums as a place to visit as much as it should.

The Stirling to Aberfoyle bus only runs every two hours Monday to Friday only, and runs on to Balfron, Milngavie (for the West Highland Way) and Glasgow. Callander is a much easier place to reach in the Trossachs- 7 days a week service, hourly in the week, two hourly on a Sunday.

So Aberfoyle is a possible day trip, but not the simplest.

By direct bus from London’s Victoria station to:
Windsor Castle bus 702 to Windsor Town Centre, Theatre Royal (1h).

I wonder if some of these timings are Rome to Rio.

I would much prefer an easy train trip from Paddington in as little as 28 monutes.

Cambridge from London’s Kings Cross station (1h 30m).

That would be a very slow train from Kings Cross to Cambridge. The non-stops which I use only take 48 or 50 minutes.

since the trip isn't until a year from now transport answers may be different then, both in Scotland and England

By direct bus from London’s Victoria station to:
Windsor Castle bus 702 to Windsor Town Centre, Theatre Royal (1h).
I wonder if some of these timings are Rome to Rio.

I found this information here: https://www.rct.uk/visit/windsor-castle. Scroll down and click on Map & Directions.

Well for the Windsor Bus (which isn't even called Windsor Express, rather the London Line)- if you click through to the actual Reading Bus (the operator) Timetables the bus actually typically takes 1 hour 15 minutes inbound, 1 hour 30 minutes outbound, a bit longer in the peaks and as little as 1 hour 6 minutes on the last late evening bus (when the castle is long closed anyway).

Yes Reading bus term it as Windsor Express but look at the Bus Livery and elsewhere on the Reading Bus website, and several other posts I have done on this forum, and it is the London Line. Until May this year it was just an un-named Greenline route until Reading Bus left the Greenline group of companies. Before then all sorts of vehicles were used, now it is a dedicated sub fleet.

I don't mind the bus- it's a different vista than the train, and very smart, brand new buses.

Never believe what an attraction tells you (so very often wrong, out of date or both. sometimes total balderdash)- go to the actual operator for the real timetable.

London to Cambridge in 1h30 would be more typical of the Greater Anglia trains from Liverpool Street (and even they usually do it in about 1hr10) or a Thameslink from London Bridge (south of the Thames). 1 hr30 from Kings Cross on Great Northern would be an all stations service, there is always a fast train which any sensible person would catch for a full distance journey.