Whoever was looking at KES for younger children, be aware that the pre-prep is on completely the opposite side of the city (west) from KES prep and senior school (east). Similarly for Monkton, the prep is at the top of Monkton, on the southern edge of Bath, and the senior school is right down at the bottom of a valley quite a bit further out, and with a nasty hill and road between.
Garden - even with £1.5m you're going to struggle to get a large garden close to the centre, e.g Bathwick or Widcombe, or even Larkhall, Lansdown. The houses tend to be 3-4 stories high, and closely packed, with really small gardens. Many beautiful £2m+ enormous central townhouses basically just have a courtyard! Even Bear Flat (slightly further out) is tightly packed Edwardian with long narrow gardens.
To get a "normal" (for anywhere else) garden, you'd need somewhere like Weston or Newbridge Hill in the west, where some of the big four storey semi-detached villas are, at £1.5m. But to get an actually "large" garden then maybe try somewhere on the outskirts like Monkton Combe.
Hills - when looking at property addresses online I recommend having the Ordnance Survey map at hand. Many addresses which look good are suddenly not so appealing when you realise the long, steep hill you'd need to climb from town, such as up Ralph Allen drive, or the steep narrow streets which are a parking nightmare in the Camden slopes.
Driving from Chippenham to KES.
Firstly, Chippenham is not fun, it's inhabited by locals who have grown up there, or incomers who need a cheap location for commuting to Bristol/Swindon, possibly London. With the budget you have it would be bonkers to go for Chippenham unless you want a large detached period house properly in the country.
Secondly, although on a quiet evening it's a 30 min drive, at rush hour it will be quite a bit longer, especially as part of the journey will be along the clogged London Road. It could be 30 minutes just on that section. You would want to test it out in rush hour to get a better idea.