Hi Art
I don't have access to 1901
I looked again at the index and it shows plenty of police officers that were born in all parts of Ireland, so that would not have been the reason for him leaving. Good thinking though

I have found a marriage for a Joseph Birchall, son of Frederick, to Mary Healey at St Wilfreds RC church, Hulme, if this is of interest let me know
Regards, Dave