January central England

Bronte Country is very easy without a car- bus from railway stations at Hebden Bridge or Keighley. While in that area you also have the city of Ripon and it's Cathedral as well.

Before moving on from East Anglia have you covered the Cathedrals at Bury St Edmunds and Norwich. While relocating you also have great Cathedrals at Peterborough and Lincoln.

Returning to central England there is a great rail rover ticket- the Heart of England Rover- with a Senior Railcard 3 days in 7 travel after 0900 costs £75.90 or 7 continuous days £99.90- https://www.railrover.org/pages/heart-of-england-rover.html

That covers an area from Leicester up to Nottingham (for Southwell Cathedral by half hourly bus) and to Matlock (on the edge of the Peak District) and thus Derby (Cathedral and Museums), also as far as Chester (Cathedral etc), Stratford on Avon, Coventry (old and new Cathedrals) and Birmingham (Cathedral noted for it's stained glass) and over to the great threesome of Cathedrals at Gloucester, Worcester and Hereford, also to Oxford, also Lichfield (another splendid Cathedral).
While over Hereford way there is an uneasy historical divide between Cathedrals. Abbeys and Minsters. Some Abbeys are the equivalent of Cathedrals in all but name- and Shrewsbury Abbey is one of those.

So far more than enough there to fill your time. If visiting Cathedrals in number like this get yourself a Pilgrims Passport for £5 from the first you visit, and have it stamped in each subsequent Cathedral.

Well, if you're looking at six days between London and Leicester, and you won't be renting a car, I would seriously think about spending your first 3 nights in Oxford, which is an absolutely wonderful city with so many things to see and do there. The Ashmolean Museum is one of the best museums I've ever been in, and it has, of course, the Bodleian library and so much incredible architecture to see. It's a lovely city, with lots of greenery and has a really lovely vibe to it. I liked it a lot. And of course, they have good public transportation in the city, so it's easy to get around.

You could then take the train up to York for the next 3 nights. York Minster is just breathtakingly lovely. I enjoyed York, although the rest of the city wasn't really my thing that much, but I did love the cathedral and found it quite beautiful. But you could easily take a few day trips outside of the city. There is a tour company called Rabbies that's based in the UK and it comes highly recommended. A lot of people on the forum have taken their tours and really like them. They have a number of locations throughout the UK, and they run a few tours out of York, so you might want to check those out.

One is called "The Heart of the Yorkshire Dales," and the other is "Tales of the Wild Yorkshire Moors and Whitby." Both are day trips, so you could base yourself in York, spend a day or so seeing the city and especially the cathedral, and then take one of these tours, or even both if you liked. https://www.rabbies.com/en-us/tours/england/from-york