Feeling the need for a degree in math

I always think those Brit rail passes are pretty pricey for the convenience offered.
Just keen in mind millions of us Brits travel by train every day for every conceivable reason
I quite like the split ticket Apps too.
I think a couple if other regulars are likely totting up options for you as I type.
Will have a look myself

Do you need Bath to Penrith in that list?

Richard, no, we do not. We will have a car for the first week, ending up in Keswick on Friday afternoon. We will drop the car at the Enterprise Rental in Penrith Saturday morning.

Just a quick look ,I used TrainSplit App and indicated I was using the Group Save fare
About £570 ,for the
London - Bath
Bath - Penrith.
Penrith- Edinburgh
Edinburgh - London.
That was just picking tickets a few weeks away,fares tend not to alter much at all during the year on average.
But some individual departure s can be noticably more expensive mere mins apart.
The train App let's you see the ranges.
Apparently we have both the most expensive fares in Europe and some of the cheapest.
I actually quizzed a guard once on my local line about why a return date to Nottingham was cheaper than the single and he said he could not explain it either ..there's the odd quirk.

Knock £90 off the above.
The Bath - Penrith ,leg is the most time consuming per mile too with changes..great choice having a car

Not entirely related...but the cafe at Carnforth,edge of the Lakes, is a famous railway station cafe .

The most cost effective option would be 2 x Two Together Railcards then you are only spending £60 on Railcards, not £90 or £120.
Heathrow to Bath- £119 for the 4 of you, on super off peak singles on the 2 Railcards- best as you can't predict flight delays- you book a specific train but can use any train after 0955.
Penrith to Edinburgh (here you know when you are travelling so can do advance)- for the 4 of you you may well find a combined fare of £29 (I can see that for September 20 now) [at fully flexible price this journey is £120.60- a split at Carlisle achieves no benefit]
Edinburgh to London (advance fares as you know your time of travel)- the four of you between £144 and £200.- the absolute maximum you can pay for 4 fully flexible singles is £229.60
So worst case that is £349, add on the passes £409= $480. (or on 3 fully flexible singles is £470 +railcards = £530= $640.
Thus you save a minimum of $180 over Britrail, maybe $350.
For Edinburgh to London and Penrith to Edinburgh you will never save with this splitting lark, and I am not convinced those are the very keenest fares from Edinburgh, but are typical ones from a 10 minute check.
Likewise we know perfectly well by now that splitting does not work. The fact that Richard got £570 off the train split app shows what is known fine well that the train splitting apps do not work.
Obviously we are too far out for September yet and for weekend fares but those are typical fares for the time and dates of travel
That only took 10 minutes.

Richard's " Not entirely related " is well worth understanding and watching - In 1944 , one of the best ( and there are many ) British films was in production toward the end of the war . Directed by David Lean and starring Celia Johnson and Trevor Howard " Brief Encounter " is a great classic about an unrequited love affair . Major portions of the film are set in a railway station . The original choice for the location was to be Watford , to the north of London , Since the station was being utilized as a departure point for children to the countryside for safety , and potential blackouts were an issue , the Filming took place at Carnforth in Lancashire .. The station has a small museum commemorating the film and does look much as it did in 1945 . ( I'll be stopping there for a visit on my way up to Scotland this coming Autumn ) Based on a story by Noel Coward , here is the complete film , it should not be missed , and have at least one full Box of tissue to hand when you do - https://youtu.be/LguRis_h1qc

Wow! Thank you so much! That would have taken me hours to work out, I really appreciate it.

Epic scone to be had there, been a few times

By the way the Brief Encounter cafe is highly recommended, at Carnforth, in the original refreshment rooms. It is open daily, even when the Museum is closed, even for Breakfast each day. They also do a very good Sunday lunch, and evening themed events from time to time.
I know this is a diversion from the OP's post.

Its a diversion , but , oh what fun ! My wife just started drooling about the scone .

It's was an entirely unrelated post ,thanks for nearly dragging it back on topic.😉

Just a note about Heathrow to Bath. There are no direct trains from Heathrow. You have to take a train (or the Tube) into London (about an hour east) and then take a train to Bath (west), so it costs you two hours of time to be roughly back where you started. In that time you would be almost all the way to Bath on the comfortable National Express coach directly from the airport.

@ Stan's information is wrong-
1. It is 30 minutes on Elizabeth Line, less on HEX, then the express passes Heathrow Junction (ie- where the branch to the airport deviates off) about 15 minutes after leaving Paddington);
2. The through ticket won't even be taking into Paddington- it is change at Hayes and Harlington, and then Reading, so is avoiding that dog leg into London anyway.

The OP has asked about train, if the OP had asked "how do I get to Bath" he would have had coach suggested. But Britrail is not valid on National Express,

That is why for this journey only an 'open' fare has been suggested, so he can be on his way ASAP. The one and only problem with the coach is it's relative infrequency.
Train is every 10 minutes or so Heathrow to Reading, every 30 from there. As an overall journey, on the route shown and ticketed train is no slower than the coach.

"@ Stan's information is wrong"

No he is not Stuart, he is correct. Stan is correctly pointing out one is doubling back and the NE coach is the best way and quickest way to get to get to Bath in the circumstances as pointed out by the OP. Why do you keep demeaning other posters?

The through ticket from Heathrow to points west (Bath, Oxford or anywhere else) works through the change at Hayes and Harlington. I actually looked it up last night and checked the ticketed route when checking the fares for the OP. Generally through tickets anywhere do not allow doubling back on route.
If you wanted to go via Paddington, then you would have to pay extra and it would be two tickes,-HEX or Elizabeth Line to Paddington then GWR out of Paddington. It could not be done as one ticket. You could do tube to Paddington but I'm not sure why you would as you have to change And It takes longer And would still cost more than the through ticket.
And it would take more time. Exactly how much more time would depend on the connection, but wouldn't be anything like 2 hours Heathrow to Heathrow Junction.
If it's 30 minutes into Paddington, you had just missed a train so had to wait 30 minutes then pass airport junction again 10 minutes later then the maximum time Heathrow to Heathrow Junction via Paddington should be around 75 minutes.

Just a note about Heathrow to Bath. There are no direct trains from Heathrow. You have to take a train (or the Tube) into London (about an hour east) and then take a train to Bath (west), so it costs you two hours of time to be roughly back where you started. In that time you would be almost all the way to Bath on the comfortable National Express coach directly from the airport.

No, the train takes us from Heathrow to Hayes/Harlington then on to Bath. Total time 2 hours. It does not go all the way west to Paddington.
The coach (our original plan) would have us sit at the airport for 3 1/2 hours, then take us to Bristol, then to Bath. Total travel time 3 hours, 50 minutes.

We don't want to sit in the airport for 3 1/2 hours, arriving in Bath at 3:20 in the afternoon, when we can take a train and arrive at 1:24 in the afternoon.

"@ Stan's information is wrong"
No he is not Stuart, he is correct. Stan is correctly pointing out one is doubling back and the NE coach is the best way and quickest way to get to get to Bath in the circumstances as pointed out by the OP. Why do you keep demeaning other posters?

The coach is not the quickest way, given our arrival time. We have no desire to get off an overnight flight and sit in the airport for almost 4 hours waiting for the coach, which then takes almost 4 hours to get to Bath (no direct options, stops in Bristol).

Not entirely related...but the cafe at Carnforth,edge of the Lakes, is a famous railway station cafe .

Hmm. We will be leaving Conwy in the morning, driving up to Keswick for the night. We will be cutting over to drive through Ambleside and then up to Keswick, rather than staying on the M6. (I assume the M6 is our best route from Conwy?) Looks like I may have to stop off for a scone.

Yes the A55 across the Welsh coast and south of Chester, then M53 and M56 to the M6, to junction 35, and come back south into Carnforth. At the traffic lights at the cross roads (after the supermarkets on your right) turn right down the hill, and you'll see Carnforth station in front of you. There is a huge station car park.
You come onto the former fast line platforms round the side of the station buildings, turn left into the subway. The Refreshment rooms are up the first (not second) ramp on your right.
The big railway yard beyond the station is the former Steamtown museum, now West Coast Railways. Among other things they run the Jacobite steam train from Fort William to Mallaig. Quite often steam excursions come through Carnforth station.
Afterwards go back the way you came. At Pine Lakes (the end of the motorway slip road) keep on the A6 through Milnthorpe. At Levens Hall (known for it's topiary) the road becomes the A591, the Kendal bypass. Don't go into Kendal, stay on the bypass. 8 miles on you reach Windermere, 4 miles further is Ambleside, 4 miles beyond Ambleside is Grasmere (Dove Cottage), and 14 miles on up Dunmail Raise and past Thirlmere (a Reservoir for Manchester) you reach Keswick.
Thirlmere now also supplies West Cumbria (where I live) with all it's water, as our supplies in Ennerdale had to be closed due to newts in the Lake (honestly)- a change in the last year.
There's a useless bit of trivia.
If you weren't calling into Carnforth you would have stayed on the motorway to Junction 36, then picked up the A591. Not much point in going back onto the motorway for 1 junction, given the length of the J35 and J36 slip roads

By the way, in the unlikely event of a rail strike you would get a National Express coach from outside Penrith Railway Station at 1.15 pm, which arrives in Edinburgh at 5.15 pm.
Although on every strike day there has been a restricted train service running from Penrith to Edinburgh, either way you wouldn't be stuck.
There is a nice park opposite Penrith Station, with a ruined castle, should you have a bit of time in hand. Also 2 supermarkets within a few minutes walk of the station, as well as McDonalds.

By the way, in the unlikely event of a rail strike you would get a National Express coach from outside Penrith Railway Station at 1.15 pm, which arrives in Edinburgh at 5.15 pm.
Although on every strike day there has been a restricted train service running from Penrith to Edinburgh, either way you wouldn't be stuck.

Oh, I do hope that doesn't happen. We will only be in Edinburgh that day and overnight, so were hoping to see the Palace of Holyrood House and walk along the Royal Mile.

So much great information on this thread, thank you all!