3 days to spare between York and Conwy

The obvious choices will be Manchester, Chester or Liverpool. ll will provide more than enough of interest for 2/3 days and are easily accessible from York and on to Chester.
The Peak District National Park is also on the way but is more difficult to access by public transport. A day trip could easily be done from Manchester though.

Thank you. I should have included that we are headed to Liverpool and Manchester after Conwy.
I've looked at Newcastle Upon Tyne (considering it) and Durham (not sure I'm fit enough for the climbs).

You could go to Sheffield or Chesterfield. Neither are regular tourist stops but both have decent access to Chatsworth House, probably the finest stately home in the country. Then a bit of time in Chester?

The Lake District, we stayed in Keswick,
or Chester

If you're in York anyway what about the North York Moors or the Yorkshire Dales, then you are later making logical progress to Manchester and later Conwy.
Find a cosy country pub on the Moors or in the Dales for those 3 days- Grassington is easily accessible in the Dales or Leyburn/Aysgarth/Hawes all with easy transit from Northallerton.

Newcastle would be a good base for 3 days. The city itself is a good place to visit. The quayside is an iconic sight with its numerous bridges, Grey Street is an example of beautiful Georgian architecture and there is the ‘new’ castle itself, which dates from Norman times, among other things. From the city or it is easy to reach the coast on the local Metro light rail. I’d recommend Tynemouth. You can also take the train into Northumberland and visit Hexham or other pretty country towns, or even see the Roman Wall.

Liverpool, Ludlow, Chester, Manchester, Newcastle upon Tyne

Thanks to all of you. We adjusted the plan a little and are going to Conwy earlier than originally planned and go spend a little time on the Isle of Man.
Always enjoy seeing all of your comments.