Which city for two days and which city for one?

Just a couple of things:-=

neither place is a city. Bibury is a village, Shrewsbury is a county town.

Bibury will share all its loveliness with you - and it is lovely - in an hour or so (probably less), maybe two if you stop for lunch (nice restaurant overlooking the stream). You cannot get to Bibury by train (or even very easily by public bus) - nearest station Cheltenham.
Bibury has one pub, one tea room, one restaurant, one trout farm, one beautiful row of cottages (Arlington Row), one river and one post office/shop. Also one Baptist church and one Church of England church. Buses occasionally go to either Northleach or Moreton-in-Marsh.

Shrewsbury is considerably larger but still only a small town. It is in Shropshire after all. The Cadfael walks are no more unfortunately, the medieval part of the town is quite small. I'd see everything I wanted to there in less than half a day unless I had to visit a prisoner in the prison (the largest building in the town). Bridgnorth isn't far from there and has both an upper town and a lower town on the Severn River (linked by an ancient funicular railway) and the world class Severn Valley steam Railway which goes for 16 miles through Shropshire and Worcestershire countryside following the Severn to Kidderminster. You cannot get directly between Shrewsbury and Bridgnorth by train except by the very circuitous route via Kidderminster (between Birmingham and Worcester).

There is no cathedral in Shrewsbury.

I went to Shrewsbury last year to watch my soccer team. Explored the town thoroughly in a morning. It's lovely.
You could use it as a base to explore the area which is very nice.

I have never heard of the village you mentioned (I am English and worked in call centers for national insurance companies and also have traveled the uk extensively with my job and my father's job) so perhaps you might want to consider somewhere else.

Since you are coming into Holyhead, you could easily get to Chester.
There is plenty to see there, including the medieval buildings you spoke of.
Warwick is also a town with many medieval buildings left, quite compact center,
with a medieval castle as a bonus. Warwick is directly on the rail line and easily reached.

If it's great cathedrals you wish to see, consider taking the train to Bath for a couple of nights, then on by train to
Salisbury for a couple of nights.
Salisbury Cathedral is magnificent.
Trains run from Salisbury to London with great frequency, if that is to be your end point.
If cathedrals are your thing, you could take the train from London to York on a day trip to see York Minster there
and the medieval town and walls.

If you are still wanting to go to Bibury, there are a couple of hotels that showed up there when I searched.
The Swan Hotel looked lovely:
https://www.cotswold-inns-hotels.co.uk/the-swan-hotel/
Nigel has given you information about the town with a rail station nearest to Bibury.
You could take a taxi from there to Bibury.

Nearby Cirencester was an ancient Roman town. An interesting town to visit with an excellent museum devoted to Roman artifacts found in the area, the Corinium Museum:https://coriniummuseum.org/

Moreton-In-Marsh is a town easily reached by rail. It is a quiet little town, part of the Cotswolds, easier to reach without a car than Bibury. It is an easy rail journey back to London from here when your time is up. There is an excellent company that does tours of the Cotswolds from Moreton, the Go Cotswolds company. They do a day trip around to a half dozen Cotswold villages.
https://www.gocotswolds.co.uk/
We took this tour last spring and highly recommend them.
There are several B&B's in Moreton that people on this forum have stayed in and liked.
Moreton would be a pleasant village in which to stay, similar (but not identical) to Bibury.

Thank you everyone. I only heard of Bibury when I found a site - 27 of the most beautiful small towns in Europe. And I thought Shrewsbury had a cathedral. I must be getting my historical novels mixed up, or it's a common old age brain fart. Wasn't that the cathedral in Pillars of the Earth? I'm too lazy to look it up again. And I love Brother Cadfael but no more tours there. So - Now I don't know where to go. We have 3 days in London, 3 days in Dublin and 3 days left before we have to return to Southampton for the ship coming home.

Any suggestions?

I was just looking at a thread you posted last fall. Did you get to go to Bath, Salisbury, Cotswolds and York, as you planned on your November trip?
Knowing you've already seen these places, I guess we don't need to suggest you see those again.
https://community.ricksteves.com/travel-forum/england/ten-days-in-the-uk-by-train

As good as Pillars of the Earth was (or is), many of these series are fiction, based loosely on some facts. Meaning you cannot take them literally sometimes. Also some of the abbeys, cathedrals and castles have disappeared since the middle ages. Henry VIII tore down many of the abbeys in the 1500's. Since then much has been destroyed by civil wars in England. Some cathedrals that used to stand were bombed in World War 2. So if Pillars of the Earth speaks of a cathedral or abbey that was in a certain place in medieval times, it may not be there now.

If you will be more specific about what you are seeking, there are lots of us here willing to help.

Are you looking for towns/cities with cathedrals?
Are you wanting a town with a medieval center?

You could just stay in London for your extra time and do day trips out to see castles such as Leeds or Hever.

Have you already seen Windsor Castle and Hampton Court Palace?

It all depends on the timings of ferries / trains - but I would consider:> Dublin to Holyhead by ferry - then train to Bangor. Now take a bus to Caernarfon - which has a wonderful castle.http://www.caernarfon-castle.co.uk
Now, depending on timings, you could stay overnight in Caernarfon or Porthmadog.
To get from Caernarfon to Porthmadog use the Welsh Highland Railway =http://www.festrail.co.uk

Next day = ‘normal’ train to Shrewsbury via the scenic (slow) Cambrian coast line. Have a look around Shrewsbury before taking another train south to Ludlow = have a look around. Like Shrewsbury, it does not look good from the station but wander into the town and you will find plenty of medieval buildings. Then, continue south by train to *Newport where you change for a train that will take you to Bath in around 45 minutes. Overnight in Bath - if you have seen Bath, consider Cardiff as an alternative. * You may wish to stop off at Hereford on the way. It does have a medieval cathedral but if it came to the crunch between 1 of 2 cathedrals on my suggested route, I would choose Salisbury - which would be on the final day - between Bath & Southampton. (It is also quite walk from Hereford station to the Cathedral).

Final day - see Cardiff / Bath before taking train to Southampton. (No need to go back to London as direct trains operate Cardiff > Newport > Bath > Southampton > Portsmouth hourly).

As an alternative to the above:> Dublin to Holyhead by ferry then train to Conwy (Conway). (Virgin Trains do not stop at Conwy but Arriva Trains Wales do - as long as you inform the guard earlier). See Conwy - which has a medieval Castle/town right by the railway station, then back to the trains for Chester (overnight). Next day - south by train to Shrewsbury & then as above.
Google the places mentioned for more info.
http://www.castlewales.com/conwy.html
Trains = www.nationalrail.co.uk
Buses = www.traveline.info

There is no doubt that Bibury - specifically Arlington Row and the river Coln - is beautiful. So beautiful that many tourists stop there for the obligatory selfie (themselves in most of the picture with a nod to the beauty behind) in their tour buses, but it is remote and not easy with a BritRail pass.

Another city near your route would be Worcester. The cathedral - and its magnificent ring of bells - is very impressive and chock full of history. A little further west is Hereford with its famous cathedral and chained library - yes the books really are chained - and incredible history. Hereford like many of the surrounding villages like Ledbury (on the train) have large areas of black-and-white ancient buildings and narrow lanes. If you like English cider, Hereford is where it comes from.

There is no cathedral in Shrewsbury.

Shrewsbury does have a cathedral. But it's not of any great architectural merit being a 19th century gothic revival Catholic Church. The great gothic cathedrals in that part of the world include Worcester, Hereford and Gloucester, perhaps you thinking of one of those?

Bibury is a village which usually has far more tourists than residents.

Do you mean Cirencester or Chichester?

If you're interested in cathedrals and medieval history and taking into accoint your departure from Southampton it may be worth considering Salisbury, Winchester and Chichester. All have impressive cathedrals and a wealth of medieval and earlier history. The proximity to Southampton and the ease of access via train makes these three cities an attractive option within your timescale.

Considering your departure from Southampton, I like JC's suggestions of Salisbury and Winchester.

For something completely different.......
you could arrive at Southampton and take the ferry boat to the Isle of Wight for 3 days.
It is a popular beach holiday destination, mentioned in the Beatles' song "When I'm Sixty-Four".
You could visit the palace built by Queen Victoria, Osborne House.
Enjoy the restaurants and walking trails around the island.
Then take the ferry boat back to Southampton for your departure.

You don't mention what month you will be making your trip.
Isle of Wight would be best done in warmer months; April through September;
but not in winter months.
Bike rentals available on the island if that is of interest.