Central Bath is mostly flat and paved — pushchairs are straightforward in the city centre. Exceptions: cobbled sections of the Abbey Churchyard are manageable but bumpy; steep pavements on the hills above the city (Lansdown, Bathwick Hill) are not pram-friendly.
Baby-changing facilities: Holburne Museum, Roman Baths, SouthGate shopping centre (multiple), Thermae Bath Spa (lobby, though no children inside), and Café Creme Kingsmead Square.
Nappy changing in the open: Victoria Park has bench areas; the Abbey Churchyard has sheltered seating. Neither has dedicated outdoor facilities.
Quiet feeding spots: Holburne Museum café, Topping & Company bookshop, and any of the above cafés. Bath Spa station has a nursing room.
For evening out without children: Bath's hotel concierges can arrange registered babysitters with notice. Check with your hotel directly.