I think the church you are looking for WAS St John's, because that was the only church in that area with a spire. Neither St Werburgh's nor the Hind Street Methodist Chapel had a spire, whereas St John's had a very tall traditional pointed spire.
St John's - which was just off Borough Road - was still standing in 1967. Around that time, clearance of much of the area was about to begin, to make way for the new shopping precinct.
It could be that scaffolding had been set up around St John's to help with demolition of the church. The tragedy of your ancestor's death would certainly have been reported in the Birkenhead News
and possibly the Liverpool Echo (there was no Wirral Globe at that time) and an inquest would have been held.