Update: Itinerary help/ suggestions

Although I enjoyed York, it's less than half the size of Edinburgh, so I would alter the distribution of nights there unless your primary intention is to hike. There's so much to see in Scotland, I feel as if it's a shame to go there just for 3 nights in Edinburgh.

Mid-July into August is peak school holiday time, so expect things to be busy and more expensive than at other times of year.

Great to see some love for Lincoln - one of England’s finest cathedrals, and there’s a great castle too. However, unless you’ve got family or other reason for staying, you can see the highlights in one or two days, so I’d add some time elsewhere as three days feels a little long.

I agree that your York/Edinburgh balance is wrong. Also, if you’re using public transport, Durham is on the way to Edinburgh so you could look at a night there en route rather than as a day trip from York.

The whole of August, Edinburgh hosts the world’s largest collection of festivals, including - but not limited to - the Fringe, the International Festival and the Tattoo. This makes August either a brilliant month to visit Edinburgh or a terrible month to visit Edinburgh, depending on your point of view: very very busy, hotels three times their normal price, loads to see and do.

I’m mentioning this so you can decide if it’s your kind of thing. If not, make sure you’re in Edinburgh at the start of your trip, in July or after 26th August.

3 days is a perfect amount of time for Lincoln. In the uphill area as well as the cathedral, castle there is the much overlooked Museum of Lincolnshire Life which is a half day visit - a rarely known fact Lincoln is the birthplace of the tank, which the museum has an original and restored WW1 tank which was built in the city. This in itself, as well as the Jews House and the Magna Carta Experience would easily take a day and half to do them justice. That is before before moving down hill to the High Bridge/Glory Hole (the only extant Tudor building on a bridge) which is now occupied Stoke's coffee shop, which one can smell the coffee grinds 100s of metres away, then one can move on to the Brayford Wharf area which was described one guide book as the Melbourne of the UK. Across the other city there is the International Bomber Command Memorial on Canwick Hill with it its spire which dominates the southern side of the City.

One advantage of Lincoln has over York, it has yet to be Harry Potterised and is more authentic and less overtly touristy. On the way from Cambridge an easy diversion is to call in at the Battle of Britain Memorial Flight visitor centre at RAF Coningsby and see the one of only two flying Avro Lancasters in the World as well Spitfires and Hurricanes owned and maintained by the Royal Airforce, the nearby Tattershall Castle is a must too, which must be everybody's idea of English castle.

We love Great Britain, especially the countryside. Not renting a car will mean that you will miss a lot of the wonderful scenic countryside.

1) York was one of our favorite cities in GB and we did three nights and days there. Don't miss the National Rail Museum, walk the ancient walls and other museums in the city. We did Durham as well and spent 2 nights there. Consider doing 5 nights in York and then moving on to Durham.
2) Keswick, we spent three nights there in late October and enjoyed it. It was a bit chilly, but we used our rental car to drive around the Lake District. How do you plan to see the Lake District without renting a car?
3) You aren't doing the Cotswolds, we loved it staying at Chipping Campden, using it as a base to visit Oxford, Blenheim Palace and Stratford Upon Avon.

You only spent three nights in Scotland, visiting only Edinburgh. Consider skipping Edinburgh this time, come back later for a more intensive visit of Scotland. We loved St. Andrews, the area between it and Edinburgh as well as Loch Ness and Inverness.

Compared to the cathedral and castle - and indeed the glories of Durham, York, Cambridge and the Lake District and Edinburgh, in my view both the Museum of Lincolnshire and the Glory Hole are somewhat minor attractions and I maintain that on a short visit to the UK, time could be better distributed.

If you do go to Tattershall Castle it is an hour from Lincoln on Brylaine's B5 bus to Boston which is mainly hourly, likewise RAF Coningsby.
Bomber Command Centre is a bus every half hour, from Lincoln.

Regards @geovagriffith comment about how do you get around the lake district without a car, that is a not well informed comment, often repeated on here by those without adequate knowledge.

If you've been in the Lakes for a few days and with a car, or on a RS tour, then you aren't going to know how easy it is by public transport.
There are several threads done by me on where public transit gets to in the Lake District.
In some ways transit is easier as you are not constantly looking for, and paying for car parks, you can do linear not circular walks. Not be coping with the sometimes narrow country roads and visit a country pub or two without worrying about being caught for DUI.

"Bomber Command Centre is a bus every half hour, from Lincoln."

Do you mean the Bomber Command Memorial on Canwick Hill (which you can walk from the centre of Lincoln) or the Battle of Britain Visitor Centre at Coningsby, which is a bus ride away?

Laughing Spam Fritter- the place on Canwick Hill is called the International Bomber Command Centre.

My father is named and has his place there at the bequest if the RAF, because of the part he played in the Cuban Missile Crisis when he was Chiefy on 617. I can assure you everything is named as International Bomber Command Memorial.

Thank you everyone for suggestions made for our itinerary. We will think about how much time to spend in Scotland and also Lincoln. I didn't mention that we have been to York before (and have not been to Lincoln yet) and loved it and will use it as a base this time to visit 2 or 3 places outside of York.
Also, we hope to take buses around the Lake District from a base in Keswick.
We are thinking about the Cotswolds possibly. We did go there this past May with our base being Broadway. We loved it!
Any thoughts on public transit in and how much time in Buxton? Is a week in Keswick more than enough time?
Thanks everyone

One thing about travel in Derbyshire is that for many people a Derbyshire Wayfarer ticket (for all trains and buses) is usually the most economical ticket (though that may be a bit skewed due to the £2 bus fare cap for 2024). However for Senior Railcard holders (no other railcards!) it certainly is. Even currently with the £2 fare, in flexibility terms, it probably still is worth it.

For Derbyshire bus maps and route timetables see here- https://www.derbyshire.gov.uk/transport-roads/public-transport/timetables/bus-and-train-timetables.aspx then click through to the relevant page.

I think 5 to 7 days is a nice time for the Lake District. It allows you to have flexibility around your walking if we have a period of "liquid sunshine". For instance there are 4 lakes with public boat services on them. If you did an activity around the boat services alone on each day that is 4 days gone. Each of those boat trips gives you really good low level walking, before you start to tackle the high peaks.
If it's wet the Windermere boats give you access to the Lake District Aquarium at Lakeside for instance, and the steam train to Haverthwaite and the Lakeland Motor Museum. (as a resident in the winter I get into the latter for £5 and 75% off the Windermere boats in the winter months).
The other thing to think about is the Mountain Goat minibus tours. Apart from their itineraries they take you to several of the smaller passes which are a more interesting drive with 1:4 gradients and hairpin bends on single track roads on the Wrynose and Hardknott passes for instance.
One of the tours takes you to the western valleys of Eskdale and Wasdale for instance.
The walk I usually recommend on arrival day in Keswick, as an evening taster, is the K2T trail- the 4 mile Keswick to Threlkeld walk along a small part of the old Workington to Penrith railway line. The bus drops you at the trail start at Threlkeld, and you end in the heart of Keswick. It is just a stunning walk.
A 7 day bus ticket for Cumbria (and to Dumfries, Newcastle, Skipton and Chester) is only £34 each currently, or £70 for a group of up to 7 people. So £5 per day per person on the £34 version. One day tickets are £12 currently/£33 for a group.