I have a similar Smiths - Enfield from the 1950-1960s, I think that they didn't have model names for their clocks but I may be wrong about the model names. There's not a name labeled on my clock. Just my opinion.
Smiths and Enfield did use model names for cases though I can't help with what this one is. A Smiths "Talma" model illustrated in Clockmaking in England & Wales has very similar hands to the ones on this clock but the position of the winding holes shows the movements are different. A picture of the movement should help in identifying more about the clock along with any markings on the dial and movement. If it is an Enfield movement it is likely to be one of the F series striking ones.