You can extend a house in Romsey, but the answer comes with a filter, and the filter is how old and how central your house is. This is a town with a deep historic core and a great deal of it is protected. There are more than two hundred listed buildings and a conservation area wrapped around the old centre. On the ordinary post-war and modern streets that make up most of the town, extending is the routine business it is anywhere. In the protected heart, it becomes a negotiation.
Listing is the sharp end. Romsey holds well over two hundred listed buildings, the overwhelming majority of them the standard grade that covers everyday historic homes, with a small handful at the higher grades reserved for the genuinely important. If your house is listed, the protection applies inside and out, and it applies to things you would never expect, from windows and doors to internal walls and fireplaces. Consent is required for alterations that would be nobody's business at all in a newer property, and the works have to be done in a way that satisfies a conservation officer rather than just a builder. It is doable, and people do it constantly, but on the heritage timescale rather than the weekend one.
The conservation area adds a second layer over the town centre. Living inside it does not freeze your house, but it does tighten the rules on the things that affect the street scene, and some of the permitted-development freedoms you would enjoy elsewhere are trimmed back or removed. Changes to the front, to the roofline, to windows and to boundary walls are the ones that tend to need a closer look, because the point of the designation is to keep the character of the old core intact. That is lovely when you are admiring it and less lovely when you want a bigger porch.
In practice, the town splits neatly. On an unlisted house outside the conservation area, much of what you might want to do sits within permitted development, and a rear extension or a loft conversion is a normal project with a normal amount of paperwork. On a listed house, or one inside the protected centre, assume you need consent, assume it takes longer, and assume the answer to "can I use modern materials" is usually no. The only safe move is to check the specific status of the address and talk to the planning department before you draw up anything, because guessing wrong here is expensive.
The verdict is middling freedom. Romsey gives most homeowners a fair amount of room to extend, and a protected minority very little without going through the heritage process first. Buy on a modern street and build away; buy something old and pretty in the centre, and you have bought the constraints along with the charm.