The 1841/51/71 would appear to all tie in, in so far as Patrick Bourke and family have same birth dates, and family members have same places of birth. The odd bit is in the 1841 in that his Mother (?) Mary appears to be 20 years old meaning that Patrick was born to her at 14 years old! Given the state of the original record it could be that it should read 26 instead?
It would appear that John and Mary Bourke(Burke) left Ireland around 1836 to go to Wales with 1-year old Patrick and then having Anne shortly after arriving in Wales.
My assumption would be that unless John had a fabulous baratone singing voice, he went to Merthyr to work in the Ironworks or mines there. The birth certificate of his daughter Anne would tell us more and hopefully confirm Mary's maiden name.
By 1851 John and daughter Anne have died/gone (looking at some history sites for Merthyr it seems to have been both a dangerous place to work and disease ridden). The family then go to London. What is interesting is that there are Ellen and Mary born in London, so what happened to John Bourke? Also Mary's age seems to now be correct in this census as it suggests that she was indeed 26 when in Wales and not 20 as originally transcribed.
I can't find any of them for sure on the 1861 so I would guess that Mary has remarried, died or gone back to Ireland.
The plot thickens...........