My G Grandparents also lived in Clifton Road and many of the houses there have also been renamed over the years. At the end of Clifton Road is the old Lloyd Street School and that now houses the Conwy Archives,
In the Archives are Street Indexes and these may help. The ones they have that may help to trace your relatives old houses are the ones for 1911, 1914, 1922, 1926 and 1929. These Indexes list the name of the "breadwinner" and will be easy to find.
I'll have a look for you when I next visit the place, unless someone on here can help you beforehand.
The Archives are open Mon - Thurs so it will be next week before I can go there.