Help with England Itinerary

Unfortunately the railway line linking Oxford and Cambridge closed in the 1960s, so you have to go via London. (No doubt Stuart will come in here with more information...)

I would question the decision to travel to BAth on Day 1 and then head to Oxford on Day2.

What time do you expect to arrive in London and when would you anticipate arriving in Bath? You may well be jet lagged . How long will you have to see Bath before heading to Oxford on Day 2. Is it worth it?

Why not head direct to Oxford on Day 1 . That would give longer there, especially as you are planning on a day tour round the Cotswolds and another day to Blenheim Palace.

Unless Day 1 is a Friday I too am unsure about Bath.
The point being that on a Saturday there are direct no change trains from Bath to Oxford.
By the fall I am very hopeful that Phase 1 of East West Rail (the old Varsity Line) will be open. First test trains have run, including a Charter, and crew training is in progress.
Thus you will be able to go Oxford to Bletchley, change, train to Bedford, bus to Cambridge including using the guided busway right into the City Centre (the old rail line, now a segregated right of way).
It will be no faster than train to Paddington, tube to Kings Cross, train (or Elizabeth Line to Farringdon, train to Cambridge, or Elizabeth Line to Liverpool Street, train to Cambridge), but you will see a lot more.

Why not head direct to Oxford on Day 1 . That would give longer there, especially as you are planning on a day tour round the Cotswolds and another day to Blenheim Palace.

I agree. Even though you are spending 3 nights in Oxford, you have barely any time there, and it's such a wonderful city to visit. I would find some way to spend a bit more time in Oxford if you possibly can.

train to Bedford, bus to Cambridge including using the guided busway right into the City Centre (the old rail line, now a segregated right of way).

a couple of small problems with doing that - the 905 bus operated by Stagecoach,
https://tiscon-maps-stagecoachbus.s3.amazonaws.com/Timetables/East/BEDFORD/BE-905-230225.pdf , doesn't call at Bedford station. It uses the bus station which Stagecoach optimistically says is only "a 10 minute walk from the railway station".

It goes via St Neots so never gets on the fastest roads. Even the A421/A428 between Bedford and Cambridge isn't that fast as it is only dual carriageway in bits, and only Busway buses with the special guiding wheels on the sides can use the Guided Busway.

Due to a number of fatal accidents on the Busway with cyclists and pedestrians the Council have been forced to lower the already fairly slow Busway speed to much slower, so it isn't particularly fast even though segregated in parts.

The fastest way will be the half hourly nonstop train from Cambridge to London Kings Cross, tube to Farringdon, Elizabeth Line to Paddington and train to Oxford.

Or hire a car and count on around 2 hours each way driving yourself and then the hassles of parking and driving in Oxford.