The linked timetable says that passengers should report to the Patten Arms Hotel, or to the Manchester Air Terminus. This latter was not Ringway airport, but was a BOAC office in the centre of Manchester, at Royal Exchange Arcade, off Deansgate, Manchester 2. That's still there as well, next to the Royal Exchange theatre.
https://maps.google.co.uk/maps?q=m2+...2,70.1,,0,3.84
This was the normal way of working in those days, check-in would be here and then a (free) bus sat outside would take passengers out to the flight at the airport. BOAC had a small fleet of these buses at their city terminals at Manchester and Glasgow (for Prestwick) as well as at overseas points.