I've not heard of Ted Pritchard, but my guess is that Hall's may very well have built frames for the trade (how it ended up at TP's), or quite simply TP had customer for a frame and happened upon a suitable one at HH's.
Many builder's built for the trade, as many shops did not actually build frames themselves, but it was common practice to put the shop's name on a frame.
Suggest rather than email HH's I would call them: might help if you need to make a purchase, even if only a cable set - might help oil the wheels.
If it is an HH I'd restore it as one.
P.S Nervex lugs by the way.
Co-founder of a V-CC section, member South Eastern Road Club and owner of - 2005 Bianchi Reparto Corse (modern), 1985 Raleigh SBDU Pro Super, 1984 Raleigh LU Corsa, 1980 Allin Stan Butler Special, 1978 Ron Cooper, 1975 Bill Philbrook tourer, 2 x very early Roberts (mid/late 1960s), 1966 Raleigh Superbe Roadster, 1964 Allin Stan Butler Special Belgique, 1951 Hobbs of Barbican S/C, 1950 Hobbs of Barbican S/C, 1947 Hetchins Super Special, 1908 Centaur Featherweight. 1988 Specialized Stumpjumper.