Tentative Tour, London, Edinburgh, John O'Grotes

[PH]

Plenty of good suggestions and choice, how many months have you got 
There are some decisions to make, for all or segments, between the best cycling and visiting the most interesting places, my touring leans towards the former, for the latter I mix folding bike and the train or coach.  Andyb1's Easterly route has a lot going for it, plenty of interest, but not as scenic (Or hilly) as up through the Pennines. If visiting cathedrals is your thing, this is worth a look:
https://www.cyclinguk.org/cathedrals-cycle-route-challenge
As with the NCN, those routes are best used as a basis, rather than strictly adhered to.
The Scottish borders, as in South of the Glasgow/Edinburgh line, is an area I know little about, but having done a couple of Audax that passed through, they're on my list and there's centuries of history due to the English/Scottish animosity. The area is scenic in a rolling countryside way rather than the wilderness of the Highlands. IMO the North West of Scotland has the most scenic cycling in the UK, even if you don't jump across to any of the islands, but the things to do off the bike are few and far between.  Other parts of Scotland have some equally good bits, there's just some less good bits between them.
Couple of other random thoughts:
if you decide to ride out of London, the NCN route along the Lee Valley, from the Olympic Park in Stratford (Not Avon), is easy riding, mostly traffic free, and gets you well clear of the metropolis. 
I use trains quite a lot, mostly with a bike, it's a mixed bag, some considerably better than others.  I've never failed to get where I'm going, but it isn't always stress free!  If taking a London train from Edinburgh be sure to book one that originates there, that's most but not all.  it just gives you more time to load and you know the bike space won't already be full.  Another vote for booking with trainline, it's one of the few online options where you can reserve a bike space.
I'd second B cereus's point about trying to cram too much in, it's a trap I sometimes fall into, but exploring somewhere rather than passing through on a strict timetable usually, for me, results in more memorable tours.
If the Audax route you originally posted is more your thing, then have a search on RWGPS, for the LEL (London-Edinburgh-London) route.

[RonS]

Google maps tells me that London to Cambridge is roughly 60 miles (sorry, I am still thinking miles, not km), that is a semi-long day on a bike.

No need to change your thinking. The UK uses Celsius for the temperature, and kilos at the grocery store, but distance is measured in miles.

So warn me now if I need to also bring the French stove as that was not in my plans.

I had no trouble finding screw top isobutane cans in small Scottish villages. The tall butane cans were also on the shelf. No need to bring the extra stove.

[in4]

Just option George: Book into a Travelodge in Luton then train into London for the galleries & museums etc. Cheaper and doable day trip.

[mickeg]

Google maps tells me that London to Cambridge is roughly 60 miles (sorry, I am still thinking miles, not km), that is a semi-long day on a bike.

No need to change your thinking. The UK uses Celsius for the temperature, and kilos at the grocery store, but distance is measured in miles.

So warn me now if I need to also bring the French stove as that was not in my plans.

I had no trouble finding screw top isobutane cans in small Scottish villages. The tall butane cans were also on the shelf. No need to bring the extra stove.

Miles?  Really?  I assumed the UK was completely SI units.  I am a retired geological engineer.  In school we had to learn all of our engineering calculations in both sets of units.  And in my day to day work, we mixed units, some things were feet and inches, some things were metric.  In my day to day thinking, for distances on land, I have a strong preference for feet and miles.  But when I am in Canada I change my bike computer and GPS to kilometers instead of miles so I am using the same units as the road signs.  But otherwise I am comfortable using both sets of units.  That said, I have no clue what one stone weighs. 

Thanks for answering my stove question.  If I have a choice for butane canisters, the tall skinny ones are usually cheaper, so that is what I buy, but I can use both interchangeably with the stove I will bring.

[JohnR]

That said, I have no clue what one stone weighs. 

My last two years of school plus university were during the main changeover from imperial to metric units. I can think in both and sometimes mix them such as 2 metres of 2" x 1" timber.  A stone is 14 lbs (pounds) ie 6.36kg. Look in the shops and you will find items which weigh 0.454kg which is 1lb expressed in kg.

Our road signs never changed to km when the country nominally went metric. The official reason was probably that it would cause too much confusion although I suspect that cost was also a consideration. Some of our old road signs are cast iron so not easily modified (see https://www.countrylife.co.uk/out-and-about/in-focus-fingerposts-and-what-they-mean-235335 for some examples). Petrol is sold in litres while draught beer is still sold in pints and half pints although bottles and cans are labeled in millilitres which are sometimes round numbers and sometimes the ml equivalent of a pint. At least we've moved on from 12 pennies in a shilling and 20 shillings in a pound (the money pound (£), not the weight pound) along with some other financial delights such as the half crown and the guinea. The new penny (which doesn't buy much these days is worth 2.4 of the old pennies).

[John Saxby]

I have no clue what one stone weighs

Quaint customs they have in Foggie Olde, eh George?  Well, of course it depends...on how big the stone is, and what type of stone it's made of, innit?  So many variables... But some Unassailable Authority once upon a time decreed that one stone = 14 lbs.

My experience sez it ain't so:  We lived in Salisbury when I was a kid, and would have picnics at Stonehenge.  I flatly refused to believe that one stone there = 14 lbs, and nothing I've seen or heard since has convinced me otherwise.

[mickeg]

That said, I have no clue what one stone weighs. 

My last two years of school plus university were during the main changeover from imperial to metric units. I can think in both and sometimes mix them such as 2 metres of 2" x 1" timber.  A stone is 14 lbs (pounds) ie 6.36kg. Look in the shops and you will find items which weigh 0.454kg which is 1lb expressed in kg.

Our road signs never changed to km when the country nominally went metric. The official reason was probably that it would cause too much confusion although I suspect that cost was also a consideration. Some of our old road signs are cast iron so not easily modified (see https://www.countrylife.co.uk/out-and-about/in-focus-fingerposts-and-what-they-mean-235335 for some examples). Petrol is sold in litres while draught beer is still sold in pints and half pints although bottles and cans are labeled in millilitres which are sometimes round numbers and sometimes the ml equivalent of a pint. At least we've moved on from 12 pennies in a shilling and 20 shillings in a pound (the money pound (£), not the weight pound) along with some other financial delights such as the half crown and the guinea. The new penny (which doesn't buy much these days is worth 2.4 of the old pennies).

I have no clue what one stone weighs

Quaint customs they have in Foggie Olde, eh George?  Well, of course it depends...on how big the stone is, and what type of stone it's made of, innit?  So many variables... But some Unassailable Authority once upon a time decreed that one stone = 14 lbs.

My experience sez it ain't so:  We lived in Salisbury when I was a kid, and would have picnics at Stonehenge.  I flatly refused to believe that one stone there = 14 lbs, and nothing I've seen or heard since has convinced me otherwise.

As a geological engineer, to me a stone weighs the same as it would weigh if you instead called it a rock.  And a rock has no standardized weight. 

A pint of stout, is that pint 20 oz or 16 oz?  In USA it is 16 oz.  But Google also tells me that ounces in USA are different than UK ounces.  I choose to be ignorant of more details than that.