Yes, and with less bureaucracy than almost anywhere with a history this old. There are only 22 listed buildings across the wider area. There are no conservation areas sitting over the housing. Over the last year alone the area saw around 1,360 planning applications, so building work here is a routine part of the landscape rather than a rare fight. If you own an ordinary house, the planning system mostly gets out of your way.
The listed buildings are worth naming as the exception rather than the rule. Of the 22, none carries the top grade one designation, two are the middle grade two star, and the remaining twenty are grade two. Owning one of those changes everything: listing means consent is required for alterations that affect the building's character, inside and out, and the older the protection the tighter the leash. That said, listed buildings are a tiny fraction of the stock in a suburb built mostly in the late twentieth century, so the odds that the house you are eyeing is one of them are low. Check the specific address, but do not assume the worst.
In practice, the everyday picture is generous. Most homes here fall outside listing and outside any conservation designation, which is the combination that leaves the standard permitted-development freedoms intact. That is the route by which many rear extensions, loft conversions and outbuildings proceed without a full planning application, subject to the usual national limits on size and position. Anything larger or more prominent still needs consent, and the neighbours and the local authority still get their say, but the starting position is permission-by-default rather than permission-by-plea. The scale of local building activity confirms it: this is an area where extensions get built, not one where they get buried in objections.
The verdict is that your freedom to extend or alter a home in Stoke Gifford is high. No conservation areas, barely any listed stock, and a planning pipeline that plainly says yes far more often than no. Unless you have bought one of the handful of protected buildings, the main constraint on your extension will be your budget and your builder, not the heritage rulebook.