Mostly, yes. The typical Avonmouth house is an ordinary Edwardian or interwar terrace with no heritage label attached, which means extending it is a normal planning question rather than a heritage crusade. There are 47 listed buildings in the area and a single conservation area over at Shirehampton, so the protected stock is real but narrow. Most homes sit outside all of it. Buy the average house and the constraints on your loft or your kitchen extension are the everyday ones, not the special ones.
Listing is where the handcuffs come out, and it lands on very few properties. Of those 47 listed buildings, one carries the top grade one status and one the middle grade two star, with the remaining 45 at grade two, the most common and least restrictive tier. If your house is one of them, near enough anything that changes its character needs listed building consent, from windows to internal walls to the colour of the front door, and the planners get a say the rest of the street never has to think about. The saving grace is arithmetic: 47 listed buildings across a suburb this size means the odds are heavily that yours is not one of them. Check the listing before you fall in love, not after the architect has drawn the plans.
The conservation area is the other thing to watch, and here it is a single one, Shirehampton. Inside a conservation area the council keeps a tighter grip on anything that alters the look of the place, so changes that would sail through unnoticed elsewhere can need permission, and some of the freedoms homeowners take for granted are trimmed back. One conservation area in the whole area is a light touch by city standards, but if your street falls within its line, treat the usual rules as suspended and ask first.
In practice that splits the housing into two camps. The large majority of Avonmouth homes enjoy the normal permitted development freedoms, the rear extensions and loft conversions and outbuildings you can often add without a full application, subject to the usual size and position limits that apply everywhere. The small minority that are listed or sit inside Shirehampton need consent for work that their neighbours would not, and pretending otherwise is how people end up unwinding a finished extension. The dividing line is heritage status, so the single most useful thing you can do before buying is find out which side of it your house is on.
The verdict is middling and freeing for most. Heritage constraints here are average, which in plain terms means most Avonmouth owners can extend and alter with the normal amount of paperwork and nobody looking over their shoulder. Confirm your house is not listed and not in Shirehampton, and the building world is largely yours.