I don't see what being professional has got to do with not wanting to move to Swindon, I'd imagine it's rather more to do with not wanting to move families etc etc.
However, I'm not going to argue this one with you, cod, I know many people who work for the trust and for the most part I consider them my proficent colleagues in the museum profession. We will just have to agree to disagree.
Was that nutty DOES live in Brum? Couldn't make head nor tail of that.
If so... you do need to travel a bit.
I've not been to the back-to-backs in Brum but I know they've incredibly busy so and of course no run-around space so may not be ideal for young kids (good outreach work though). I'd avoid Wightwick in Wolves as it's entry on timed ticket (although it is amazing inside).
In worcs, Hanbury Hall has nice small garden to explore, big flat bowling green to run on
Croome Park is interesting 'cos they are restoring it, nice (about a mile) circular walk with lake and a small grotto and folly
Biddulph grange in Stoke is great, fish pond, chinese style garden, egyptian pyramid to explore
Shugborough in Staffs I've never got on with, not sure why, nice ice cream in courtyard, though
Carding Mill Valley at Long Mynd (shrops) is really beautiful, little stream to paddle in, nice easy walk up the hill to great views (no house, just free parking)
Dudmaston (bridgnorth) we go to a lot, fantastic picnic spot, lovely gardens on hill down to lake, house manageable
Attingham (shrewsbury) enormous grounds and great little kids farm (which you have to pay for as it run by tenant not NT), house good for older kids, great 'below stairs'
never been to any in Warwicks I'm afrraid which are probably closest to you. Charlecote and Upton have good reputation.