For most houses in Bishop's Waltham, yes, you can extend, and the process is the ordinary kind. The catch is the historic core, where roughly 125 listed buildings and a conservation area rewrite the rules. Almost all of those listed buildings are grade two, with a handful at the higher grade two star and none in the top grade one bracket. A pair of scheduled monuments sits in the middle of it too. Which answer applies to you depends almost entirely on which street you end up buying on.
Take the listed buildings first, because they carry the tightest controls. If your home is listed, altering it is not a matter of permitted development, and that holds for the inside as much as the outside. New windows, internal walls, a rear extension, sometimes even the choice of a satellite dish, can all need listed building consent, and the bar is set high on purpose, because the point of listing is to keep the building recognisably itself. The reassurance is scale. Those roughly 125 listed entries are a small slice of a town of a few thousand homes, and most of them cluster in the oldest streets. The bulk of the housing stock is far more recent and carries none of this, so unless you are buying into the old part, listing is somebody else's constraint.
The conservation area is the wider net. Bishop's Waltham has one, wrapped around the historic centre, and it works differently from listing. It does not freeze individual houses so much as protect the look and grain of the whole area, which means extra consent can be needed for things that would be waved through elsewhere, from demolition and larger alterations to work on trees. Conservation-area status can also trim back some of the permitted-development freedoms a homeowner would otherwise take for granted, so the safe assumption inside the boundary is that more of your plans cross a planning officer's desk, not fewer.
In practical terms, it comes down to where the line falls. Buy one of the many unlisted, post-war or modern houses outside the conservation area and a standard extension is a standard job, with permitted development covering a good deal of it and a straightforward application handling the rest. Buy a period house in the protected centre and you are in a different world: consent for more than you expect, longer timescales, and a firm steer toward sympathetic materials and detailing. Neither is a reason to walk away. Both are a reason to check the designation of a specific address before you commit to a house you intend to change, rather than after the keys are yours.
The verdict is a middling three out of five, and it is fair. Bishop's Waltham gives most homeowners ordinary freedom to extend, and a protected minority a good deal less, so the honest answer is that your hands are as tied as the postcode you choose. Pick your street with the extension already in mind and the score looks after itself.