For British-focused subjects look for any book by Simon Lavington; most likely you'll find it in a British bookstore ... for which you appear to be suitably located :->. Here in the USA they are hard to find. Three possibilities:
- Alan Turing and His Contemporaries: Building the World's First Computers
- Early British Computers: The Story of Vintage Computers and the People Who Built Them
- The Pegasus Story: A History of a Vintage British Computer
Some British-focused books from other authors:
- A Computer Called LEO: Lyons Teashops and the World's First Office Computer -- Ferry, Georgina
- LEO: The Incredible Story of the World's First Business Computer -- Caminer, David & Aris, John & Hermon, Peter & Land, Frank
- The MU5 Computer System -- Norris, D. & Ibbett, R.N.
- Calculating Instruments and Machines -- Hartree, Douglas R.
I personally enjoyed:
- Memoirs of a Computer Pioneer -- Wilkes, Maurice
(I also enjoyed both LEO books!)
Paul Cerruzi is a well-known historian of the era, but not British-focused. Two examples:
- Reckoners: The Prehistory of the Digital Computer, from Relays to the Stored Program Concept, 1935-1945
- A History of Modern Computing
There are all generally "low technical" but much, much better than a quick gloss. What it sounds like you are looking for?