For most houses in Cheddar, yes, you can extend without much drama. The heritage that makes the village famous is concentrated in the old core, not spread across the modern estates where most people actually live. There are about thirty-five listed buildings and a single conservation area covering the historic centre. The bulk of the housing stock went up between the early 1980s and the early 2000s, well clear of all that. Your freedom to build comes down to which age of house you bought.
Listing is the hard constraint, and it is narrow here. Around thirty-five buildings in the area carry listed status, and the vast majority, more than thirty of them, sit at Grade two, the everyday tier. One is the top Grade one and a couple are the higher Grade two-star. If you buy one of those, extending or even swapping a window turns into a matter of listed building consent, and the answer is often a slow yes or a flat no rather than a free hand. Listing protects the fabric of the building, inside as well as out, which regularly catches owners who assume the rules stop at the front wall.
The conservation area lays a lighter layer over the historic centre. Inside its boundary, the ordinary permitted development freedoms are trimmed back, so things you could do without asking elsewhere, such as certain demolition, larger alterations or work to trees, can need consent first. It is not a ban on change, it is a filter, and it exists to keep the old core looking like the old core rather than a row of individually improved houses that no longer match.
For the average buyer on a newer street, the practical picture is unremarkable. A modern house sitting outside the conservation area and off the listed register carries the usual permitted development rights, so a rear extension, a loft conversion or a conservatory is the standard application-and-approval routine, not a heritage fight. The village plainly builds. Something like thirty planning applications came through in the recent window, about half of them in the last year alone, which is the normal churn of extensions and alterations rather than the sign of a place where change has stalled.
So the honest answer is a middling one: average freedom, cleanly split down the middle. Buy a modern house and you can extend much as you like within the usual rules. Buy a listed cottage or a place in the old centre and you are building to someone else's brief at someone else's pace. Check the listing status and the conservation boundary before you fall for the beams.