When i found those census i looked for an English marriage of a John Lavery and a Catherine ( wifes name on census) four came up only two in Liverpool 1875 and 1882. Then i looked on SP and just put John Lavery female forename Catherine no year no place and 1 came up
Then double checking the 1871 census this John and Catherine and one of their children was born in Scotland
John Lavery 39 b Scotland
Catherine Lavery 38 b Scotland
John Lavery 6 b Scotland
Isabella Lavery 3 b Liverpool
Robert Lavery 5 months b Manchester
Then the 1891 census says John Lavery born Wigton Scotland - so was Catherine. I dont know where Wigton is or if this sheds any more light on him..............
Tracey
Scotland - Donaldson / Moggach / Shaw / Geddes / Sim / Gray / Mackie / Richards / Joel / Coull / Mckimmie / Panton / McGregor
Ireland and Scotland - Casey / McDade / Phillips / McCandle / Dinely / Comaskey + various spellings