Edinburgh to Keswick and Stow on Wold

From Edinburgh to Keswick is easy- direct train to Penrith every hour- alternates between Transpennine and Avanti trains.
Then catch the X4 (every even hour) or X5 (every odd hour) at 20 minutes past each hour from straight outside the station to Keswick- journey time 40 minutes- pay on the bus.

You will wish that you had more time. Driving from Edinburgh on the way to Keswick, worthy side trips would be Dumfries, Robbie Burns Country, and Haltwhistle and Hadrian's Wall. Keswick to Stow-on the Wold is quite a ways. If you had time you could go to Liverpool and Chester. You may be able to take a direct train to London and leave your car in Chester/Liverpool. Leave Southern England for your next trip.

Keswick to Stow on the Wold is a different matter- as Stow only has one regular bus service- the 801

That is relatively limited for exploring the area- and a car would make it easier.

But as you have asked about the trains- go back to Penrith, catch an Avanti train hourly to Birmingham. Change there for a train to Worcestershire Parkway Low Level, change there onto the High Level platform, train to Moreton in Marsh, then the bus above.
You need to ticket this in 3 parts - Penrith to Birmingham (advance tickets from around £21), Birmingham to Worcestershire Parkway £10.10 buy on the day, WP to Moreton £16 buy on the day.
The through fare, valid on any train, which you can buy on the day is £85.80. There is no current advantage to buying the through fare in advance although that might change in mid June.

This is a complicated route, but is the only sensible one. While the 801 starts from Cheltenham, it is a good walk between rail and bus stations there.

If you left Keswick after breakfast you would get to Moreton at 1650 and Stow at 1800.

Your best arrival time at Moreton is at 1530.

It's about a 5 hour drive.

To get to Heathrow it would be bus back to Moreton, train to Oxford, then the best route (although it can be done by train) would be the direct Oxford to Heathrow Coach. Compared to a 90 minute to 2 hour drive on a reasonably good day.

The car is expensive because it's flexible and takes you to the door - but it should be well below £100 a day, even for Edinburgh pick up and Heathrow drop off. Just ran a random set of days in June and first one was £320.

And you're going to struggle to fully explore without one.

Your bus and train fares (and the Penrith to Cotswolds one isn't the easiest) will be somewhere around £300

By the way the most you can pay for the Edinburgh to Penrith train is £37.40 (you should always pay less on an advance fare), and from Moreton to Heathrow is £50, so that is a total of about £180 for the three trains, rather than £300, but you should get them for a combined total of around £120 with some advance booking.

I don't see any indication of when you'll take the train trip, but you might like to know of the ongoing and sometimes frequent train strikes across the country. Not trains operated by Scotrail but cross border operators from England (Northern, TransPennine Express, Avanti West Coast among the 14 to 16 operators in most strikes) are affected. As is every train operator in the Lake District (except the little tourist train at the south end of Windermere). And trains in and around Birmingham.

So one to watch.

I believe, unless they have all ended recently (I was there yesterday) that Stow on the Wold is served by more than one bus. But they can be thin on the ground to serve surrounding areas.

All in all I'd stick with the car for the whole journey.

In my opinion a car is very helpful in the Lake District - if you will there long enough and want to park at National Trust car parks (many or most of the rural ones are) you will be well served to get a National Trust membership (with that the carparks are free, and they are a very good charity - long time active member here - and have some excellent properties in the Lake District including Hilltop).

The drive from the northwest to the Cotswolds isn't great, especially on the M6, but there are more top grade National Trust properties on the way such as Little Moreton Hall, the Beatles boyhood homes, Biddulph Grange Garden (where else can you enter a building in one country and leave it into another in a completely different continent?), Lyme (Pride and Prejudice, Mr Darcy) among many others.

The Cotswolds is just so much easier with a car - really it is. And a National Trust card could pay off there too with Snowshill, the magnificent world class gardens of Hidcote Manor Garden and many more in the Cotswolds.

You'll just wish you had a few more weeks.......

Whatever you choose I wish you well with the lovely holiday you have planned.

By the way, apart from the Lakeside and Haverthwaite Railway (at the south end of Windermere), there are six other tourist railways in Cumbria- the 'Ratty' (the Ravenglass and Eskdale Railway), the Threlkeld Quarry Railway, the Millerbeck Light Railway, the South Tynedale Railway at Alston, the Eden Valley Railway (at Warcop) and the South Stainmore Railway (at Kirkby Stephen).

Oh and I did look up the bus services in Stow before posting. There are 10 other services which serve Stow- 6 are one day a week Community Minibus services for shopping use which give minimal time at their destination (a bit like the Fellrunner bus in the Lake District- that is open for public use but almost impossible to use for trips into the countryside) and three are school services, which leaves only Pulhams 802 to Kingham Railway Station, another fairly restricted service.

It is difficult because I, who lives here, think a car is a liability in the Lake District especially in high summer when there is very limited and expensive parking, and Keswick is a real pinch point unless you park well away from where you are staying at either the Theatre by the Lake or at the old Railway Station. Both the County Council and The National Park are also trying what they can to replace car use with public transport, but that the car is very beneficial in the Cotswolds, especially where you are going.
The whole parking problem in the Lake District is chronic, and it is only time before there is a major incident which emergency services can't reach due to bad parking. I saw that nearly happen last year on National Trust property where the fire service struggled to reach a fellside fire.
Many people here have a love hate relationship with the National Trust.

On a strike day you will get to Cumbria from Edinburgh as Scotrail run to Carlisle from Glasgow. There is then a direct bus from Carlisle to Keswick. There is also a National Express coach from Edinburgh to Penrith and 2 coaches to Carlisle.

The answer could be to rent a car at Penrith for your journey south