You asked
I would like copies of postwar Administrative Areas maps or plans covering the West Midlands conurbation.
We said
Thank you for your request.
We do not hold the maps you are looking for. Maps tailored specifically for your purpose, geography and date are unlikely to exist.
However, we do have four suggestions for resources which may assist you:
1. The National Archives
Under the catalogue reference AT 98 (https://discovery.nationalarchives.gov.uk/results/r?_q=AT+98), the National Archives hold 134x county-level maps for 1951 and 133x for 1961. Note that while they include District boundaries, none include boundaries for wards. Some of these map sheets may have been digitized, but TNA will charge for supplying them.
For example, AT98/985: WARWICKSHIRE. Administrative county, county and municipal boroughs, urban and rural districts, and civil parishes, revised to 1 October 1960. Parliamentary constituencies revised to 1 October 1960. Annotated. Map scale: 126720
2. A Vision of Britain (AVoB)
AVoB, the public front end to the long-running GB Historical GIS Project, holds many County maps, mainly digitized and downloadable.
https://www.visionofbritain.org.uk/data/
Maps can be viewed, and boundaries downloaded from the AVoB website. You should find district, and parish boundaries, but wards do not appear available. You will have to register and ask AVoB for permission to download boundaries for any lower level geographies.
3. National Library of Scotland
National Library of Scotland: https://maps.nls.uk/geo/explore/#zoom=12&lat=52.52098&lon=-2.04267&layers=1&b=1
1:10,560 OS map sheets offer the historically most reliable set, but as well as the sheets' date of publication, you will need to be aware of the Surveyed and Revised dates - which may vary substantially across a sheet, and differ widely from the date of publication.
4. Academic institutions
For Greater Manchester (not your area, but similar may apply elsewhere), there are several hundred ward-level (unscanned, paper) maps in the University of Manchester's Map Library. It is possible that a similar collection may be available at the University of Birmingham:
https://www.birmingham.ac.uk/schools/gees/facilities/map-library/map-collection.aspx
Further collections may be located at other academic hubs in the West Midlands.