This is a late edition William Cook Round Badge, which carries the signature of Wlliam Cook Senior.
The cue has an ash shaft with a four pointed, hand-spliced ebony butt. The badge is ivory.
The cue was made by Burroughes & Watts, probably between 1900 and the mid 1930s.
I have a September 1931 Burwat Monthly Price List, issued by Burroughes & Watts Limited in Toronto, Canada. This price list contains a picture of the William Cook Round Badge, offered for sale under the description "Club Cue".