What street was next to billingtons chip shop on the corner oflansdown road?

Do you mean off Lansdowne Road?
https://www.wikiwirral.co.uk/forums...re-laird-street-memories.html#Post345631
Bert is your man to find the name of the road.

33 Lansdowne Rd, Mrs Agnes Billington, Fish Fryer.

Corner with Connaught St, 1938 Directory.