The CSO makes Electoral Division and LEA data available on Geohive. The maps aren't very good, though. This is a better, combined map.
Toggle between ED and LEA polygons by pressing the button. You can find the area(s) in which you're located by pressing the "locate me" button. You search for a specific place and see its ED and LEA information by using the search field on the map.
As of February 2024, these are the most up-to-date data, refreshed in November 2023.
CSO Local Electoral Areas, 2022
Some polygons have been simplified at the 1 metre level for display purposes. This should have no practical impact on the determination of the ED or LEA in which a given point is contained. If you are concerned about the accuracy of the data on the basis of your own analysis, please get in touch.
Open an issue. No guarantees, though: we're all busy.