Hi ART
Bit difficult to say for certain they are the same ones

1871 census, Manchester [St George's district] - 6 Samuel Street
RG 10/4048 folio 131 pg 24
Joseph Birchall; head; mar; 26; police officer; Ireland
Ann do; wife; mar; 29; winder cotton; Ireland
William do; nephew; 12; roller? cotton; Cheshire, Stockport
If this is him, there are good records available re the Police recruits
I don't see a Joseph Birchall on the records held by Lancashire Records office but if he was based in Manchester you could repost & ask for a lookup on the MLFHS CD of Manchester Police recruits.
Don't know if the William Birchall nephew ties up anywhere?
Also a possible marriage:
Joseph Birchall to Ann Kiloren - Manchester Registry office or attended [ie could be a Catholic marriage] 1865 Q3 vol 8d pg 699
Good luck
SandraC