Not a design I recognise, sadly. In terms of finish, they do look very much like the earlier Walter Gilbert panels made byWalsh (e.g. Labours of Hercules, Egyptian Stork, Squirrel and Pine Cones, etc.); however, I'd expect Walsh production to be marked. I don't have the Reynolds book, but it might be worth looking there for something similar, or looking at the pattern names (no pictures) recorded in Phillip Medhurst's Works Arranged and Executed byWalter Gilbert to see if there's a possible match.
If they're not Walsh, then Jobling could be a possibility. They made moulded panels: e.g. the Alan Howes designed patterns. Baker & Crowe also state that Chance made similar panels, butI've seen no further documentation on that: maybe David Encill would know more?
Hailwood Ackroyd is another contender, but the few pieces of theirs I've seen have been marked. Continental producers are also possible of course. I can confirm that this design is not shown in Decelle's Sabino Catalogue Raisonné.
Seems panels such as this are hard to identify. I have a couple of moulded panels which look very like some of the more stylised Gilbert/ Walsh lighting panels, but as yet I've been unable to positively identify either.
I'm also interested to hear about Tutty's. I grew up on the Wirral and spent a lot of time in Birkenhead, but I don't remember the store at all.