The first transistor radio seen in the UK was a battery job made by GEC, described as 'experimental', and shown at the National Radio Exhibition in London in 1954. The next was the first British all-transistor radio, the Model 710 made by PAM, a subsidiary of Pye Radio, launched in March 1956. Development was slowed by the arrival of VHF FM, which many older listeners thought 'too crisp' ... I agreed at the time, the old valve-in-wood radios had a mellow tone which I still prefer.
However, the time wasn't wasted, as manufacturers were working hard on higher-power AF amp circuits, as well as reduced distortion. The main change between '56 and '59 was the development of hybrid valve/transistor TV receivers, especially the new UHF models - radio remained more popular than the pundits thought it would, but the 'big money' in the laboratory and the shop was all going on TV and stereo equipment.
The Pye 'Slim Six' mantelpiece receiver, dated 1958, doesn't give any details about AC or DC, but although it's obviously designed as a domestic item (not a portable), I think it must have been battery.
There was a rush of AC/DC models in '58 and '59 (details if you want), but many of the most popular were like the Bush TR 91, a 7-transistor battery table model - quite elegant, and quite good performance. The manufacturers were fascinated by the reduced heat and increased circuit safety, as well as the fact that 'blown valves' stopped being a problem, there was no hum, and the radio could easily be moved around the home.
What all this is leading to is that I think you'll be lucky to find an early AC-only transistor job, although there were a few hybrid valve/transistor radios which were AC-only in the late fifties.
Hope this is at least interesting. If you're a historian, try 'Radio, Radio', written by Jonathan Hill, published in the UK by Sunrise Press, Bampton, Devon, in 1986.