Yes, in most of Sandhurst you can, and the paperwork will cause you more trouble than the heritage rules. Only a handful of buildings are listed. There's one conservation area nearby, and it isn't where most people live. The neighbours put in planning applications constantly. The obstacle is more likely to be the council's forms than a heritage officer.
There are 31 listed buildings in and around Sandhurst. None is Grade I, 2 are Grade II* and the other 29 are Grade II. Listing covers the whole building, inside and out, so in a listed house even replacing windows or taking out an internal wall needs listed building consent on top of any planning permission. Works done without consent are a criminal offence, not a telling-off. With 31 listed buildings in a town of post-war estates, though, the odds that you're buying one are low, and if you are, the particulars will say so loudly.
There's one conservation area in the surrounding area: the Royal Military Academy's former Staff College and London Road, on the Camberley side. Inside a conservation area, some changes that would otherwise go ahead without an application need consent, and demolition and work to trees come under closer control. For the closes of Owlsmoor, College Town and Little Sandhurst, none of that applies. There's nothing in the record about extra local restrictions beyond this, so check with Bracknell Forest before assuming either way.
The neighbours aren't holding back. Between April 2024 and July 2026, 2,466 planning applications were lodged across the area, 1,008 of them in the last twelve months. That tells you nothing about which were approved. It does show that extending, converting and rebuilding is a routine part of life here. A loft conversion in Sandhurst doesn't cause a scandal. It gets you a neighbour asking for your builder's number.
For most houses outside the conservation area and not on the listed register, a lot of modest work falls under permitted development rights, which allow certain extensions and alterations without a full planning application, within size and position limits. Anything bigger, closer to a boundary or higher than those limits needs permission. Before you rely on permitted development, check whether the house has lost those rights through an earlier planning condition, because buyers who skip that check often find out from an enforcement letter. Getting a lawful development certificate costs less than taking a wall down.
Your freedom to build here is reasonable but not unlimited: there's rarely a heritage barrier, but the ordinary planning rules still apply, so check them before you buy the house for the extension you're already picturing.