Yes, and your neighbours have probably already done it. Only a few dozen buildings in the area are listed. No conservation area is recorded. Hundreds of planning applications go in every year. What will cost you is the clay under the house, not the planners.
The wider area holds 44 listed buildings: one at Grade I, one at Grade II* and 42 at Grade II. Owning one changes everything. Listed building consent is needed for alterations that affect the building's character, inside or out, and that can include windows, walls and fixtures the owner might assume are theirs to change. For everyone else, and that is almost everyone in a village of post-war estates, the listing register is somebody else's problem. The mapping also records no conservation areas and no scheduled monuments inside the area, so there is no blanket heritage control over ordinary streets.
Between April 2024 and June 2026, 1,095 planning applications were made in the area, 492 of them in the last twelve months of that period. That is steady evidence that people here extend, convert and rebuild as a matter of routine. It tells you nothing about how many were approved, so do not read it as a promise. It does mean the council has seen plenty of loft and rear-extension plans in Twyford before yours.
The ground needs more attention than the paperwork. Every postcode in Twyford is rated class 3 for shrink-swell on the British Geological Survey's published hazard scale, meaning clay that swells when wet and shrinks when dry. It is common across the Thames Valley and no reason to panic, but it does affect foundations. An extension on this ground may need deeper footings than a builder's standard quote allows for, especially near large trees. Get a structural engineer's view before you sign off the budget, not once the digger has found the problem.
In practice, most homes that are not listed can take a modest rear extension or loft conversion under permitted development rights, within size limits, without a full planning application. Anything larger, anything at the front, and any work on a listed building needs consent from Wokingham Borough Council. Check which route your plans fall under before you draw them. Flats and some newer estates may have had those rights restricted, so the deeds and the council are worth a phone call first.
Freedom to extend is one of Twyford's real strengths. Planning rules will rarely stop you. Budget for foundations to suit the clay.