I’ve said it before and I’ll say it again - Bowes Museum is absolutely diabolically naff for a young child. In child terms, it’s a posh house full of big rooms, covered in paintings or cases of decorative ornaments like vases. There’s a slide far in the grounds if you venture off a bit and a wooden platform to climb on if I remember rightly, but not a proper park.
Plus once you’ve been to Plotters Forest once, you really don’t need to go again for a very long time. It’s just an outdoor play area which isn’t very big either and not a chance Bex could actually interact with Nina on it; she’d be stuck on the accessible wooden platform which takes about 5 minutes to walk the length of. Far better things much closer to her house!