So Cheshire and other areas surrounding Manc are lovely (we have good friends in lots of the areas mentioned) but I am going to risk getting flamed here and say there is a high percentage of a certain type of person that populates these areas and when we were in the same position as you we discounted these areas for that reason 😬.
Also you need to have a think about what you would like/what do you need for your 550k because it will vary greatly - it won’t go very far in Cheshire.
The Wirral is a great shout, I love Liverpool such a nice city. I’m from Leeds and find both Leeds and Manc very aggy.
York also great shout, fantastic state schools and lots of lovely suburbs that have a community in their own rights but are walkable into town.
I’d definitely consider the pretty market towns surrounding Harrogate, so in order of personal preference Knaresborough, Wetherby, Boroughbridge and Ripon, lovely and enough there in their own rights but short drives or train journeys to Harrogate or York and close enough to Leeds for a bigger city day trip either on the train or driving, also all near stunning countryside and brilliant kids attractions. You’ll also find smaller villages surrounding all these towns which are very lovely.
Stocksbridge and Dore in Sheffield are nice - good schools, Stocksbridge is more like a small town and Dore a big village but both lovey and close to countryside. Sheffield’s a pretty underrated city but I really like it’s got some cool areas because of the Uni.