Although I'm 90% sure that the Telegraph Inn/Hotel became the Bidston Hotel, I'm slightly confused over roads. Most later 19c maps have what is now Hoylake Road as Corporation Road and have yet to find a map with that part marked as Bidston Road as per bert1's posts. However have looked at Brooms Plan of the Borough of Birkenhead and suburbs 1888 which Mark kindly posted on wiki for downloading which has Bidston New Road (think it became Vyner Road North) on it
I agree, Derek, that's Vyner Rd. North now. Interestingly, on the 1912 O.S. map, both Vyner Rd. N. and S. are simply named Vyner Rd. Oh, on that map, Hoylake Rd. is marked.
Interesting to see on Broome's Plan that it looks as if Mr. Vyner had intended to build houses on quite a lot of Bidston Hill, but in the end, only Eleanor Rd. was built, and no houses on the hill side of that road, only on the Western side, so most of the woodland was preserved.