From Bath to York you could visit Stratford Upon Avon for some Shakespeare, then a short hop onto Naseby for some English civil war sites.
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The UK has loads of Cyclosportiv type events. Often based at the many national parks and all the favourite routes including those roads used for national events. Event calendar below.
Iāve been thinking about doing something similar. Am wondering what you will do for a bike? Take your own, buy a new one, to bring home or flog while there, Hire if so who from, etc? How are you going to transport it, campervan?
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Bath to York has a few options. You would take the M5 motorway towards Birmingham (possibly stop off at Stratford for Shakespeare?) and then a choice, either M1 direct North towards York/Leeds or M6 towards Manchester and then the M62 over the Pennines. M6 option could include a detour to Ironbridge perhaps, broadly half way.
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For the full ancient history tick Iād add a stop at the Uffington white horse on your way from London to Bath . The views north across the Cotswolds from the top of the hill are quintessentially āenglishā
White Horse Hill | Oxfordshire | National Trust
Between Bath and York Iād suggest a detour to Chatsworth house for Afternoon Tea and a chance to stretch your legs around the stunning formal gardens.
Chatsworth
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This is ambitious but I think you know that by now - seems like quite a few others have warned against trying to do too much so I wonāt frame what Iām saying as a suggestion/warning. What I will say is that the abundance of warnings doesnāt surprise me. Itās almost a predictable trope that Americans visiting here will try to take on way more than they have the capacity to really fully experience.
As long as you know and understand what youāre getting then donāt let any of us scare you off of your dream itinerary though. I think weāve all just seen lots of visitors commit to the whirlwind version and then leave wishing theyād spent more time on fewer places. But if whistlestop is your style and youāre going into it with your eyes open then donāt let us stop you! 
Iām from California and have been in Glasgow for 15 years now so I see those travel patterns quite a lot from friends and family (and I did the same too when I first arrived!) I think the thing that @splash tried to highlight which is really difficult for us Americans (especially west-coasters) to comprehend is that sizes/distances/etc. just donāt scale in the way we expect. Best example I can think of in California is comparing San Francisco to Los Angeles - because SF is a Victorian pre-car city the amount of history/variety/travel-time packed into each square mile is just way more dense than LA.
But the UK (and Europe) is an even more extreme version of that difference - most of the civilization on this island had been continuously inhabited for over a thousand years even before San Francisco got started, which means the scale of distances here isnāt just pre-car but also pre-industrial-revolution. Thatās how you end up with accents that are so wildly different across distances of as little as 20 miles - anything further than a dayās walk for a horse and cart meant people just didnāt interact with each other often enough for their speech patterns to align (which also meant theyād end up with complete localised histories/cultures/artifacts too).
And to take it even further - a huge amount of that highly localised history and culture was already in place pre-printing-press. So even if the East Coast of the US had a thousand years to establish before the industrial revolution instead of a hundred, it would still have ended up more homogenous than anywhere in Europe just because of the efficiency of how information could be distributed across larger distances.
So if you have to zoom in 2x to get from LA to SF in terms of scale, youāre going another 2x to get to the East Coast and then another 2x again to get to the āold worldā. (Obviously those are highly exact and incredibly scientific numbers that are definitely 100% accurate 
You get the idea though.)
ā
Sorry, ended up on a bit of a tangent there! Iām keeping it because I think even more than informing how I might plan an itinerary I find that lens really helpful just in terms of my understanding of what Iām seeing and experiencing regardless of how much or how little time I choose to spend in a place. 
If I were to make one specific suggestion Iād maybe say to leave that second āScotlandā day undefined til you get here if you can bear to do so - you might find you just want to stay an extra day in Edinburgh and leave the rest of Scotland for another visit when youāve got time to spend a few weeks just exploring north of the border. Enjoy your trip, wherever you end up spending your time!
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Digressing, but if youāve not been youāll have to go to the California which is circa 19miles north east of Glasgow, some good cycling roads round there
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anniversary trip, my wife doesnāt ride and not planning on throwing my leg over the bar of a bike unless its easy/cheap and more convenient than taking a bus/metro.
Yes I get it. The sticking point for many is āfull or deeper experience.ā Iāve given the analogy that we are looking for a smorgasbord, or in Spain we are going for tapas instead of a full meal. We arenāt planning to move to the UK, this isnāt a where should we live trip (we are doing those right now in the US). Itās a deliberate and conscious choice. My wife wants to make āthe one tripā to the UK & Ireland and sample a subset of the most iconic and important historical places.
As an experienced car traveler, the Bristol-to-York drive looks unappealing and nothing of particular interest to see along the way (other than Conwy/Snowdonia which is out of the way). And it is only $70 (x2 = $140) to fly from Bristol to Edinburgh. So Iām thinking of flying from Bristol to Edinburgh, hire another car, skip York, and do Vindolanda as one of the day trips from Scotland.
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There are lots of interesting places between those two cities (obviously - youāre talking about driving through half of England), but youāre not going to see them from the motorway.
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Sure, thatās why I said of particular interest (to us).
FWIWā¦
Before posting on the forum, I reviewed the key info spread over 1800 pages in these 2 books:
and started with the red Must See list in the front of the GB book:
and we watched a collection of YouTube travel videos.
Then I looked at his suggested car trip:
Looks interesting, but aināt nobody got time for that except the retired!
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Its a not bad car trip he suggests but you actually miss out quite a bit! Enjoy what you do see and maybe by autumn (fall) the UK summer will have actually arrived 
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Read about the Tor, and wanted to ring up Peter Gabriel and have him write a song about it! If you arenāt old enough or familiarā¦ Solsbury Hill (song) - Wikipedia
yes this one:
made me think of Peter Gabriel and nearby Solsbury Hill.
no, Iām captain random 
keep up, would ya? LOL.
As someone already mentioned above, I think a good choice of pubs will highly increase your enjoyment. Itās a big part of the culture.
Many pubs are listed buildings full of history, and if they have a carpet, you can even smell it!!!
The British desSerts are amazing!
If heās going to include Windsor, he could at least try and put it in the right place. 
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UK summer = Spanish Winter 
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The desserts arenāt bad either
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Good eye! I had a boss in Cambridge for a couple years, and noticed that too! And Iām not going to Arrakis looking for spice, so desserts for me too!
Lol good catx!