Thank you all for your quick and helpful reply. I was away last evening, so I could not answer the queries. To answer them now:
--No, I could not find the couple in the 1841 census
--Yes, I did find the father and some children in 1871: RG10 3768 93 6.
I looked for a marriage of Daniel Birmingham in Ireland (on RootsIreland.ie) but did not find any fitting marriage prior to 1839.
The birth and baptism of William Bermingham found by rosie99 is very much appreciated. William is the person in my family tree I am especially interested in.
The Anne of 1847 also found by rosie99 does not appear in the 1851 census. However, she makes the death of Ann in Dec 1847 more likely the death of Daniel's first wife.
I still have a problem with the early children (of Anne and Daniel): I cannot find them in the GRO records (which would give me the mother's maiden name).
For convenience, the children are:
[From Daniel and Anne:]
James born circa 1834 Ireland (census 1851)
William 04 Aug 1841 Liverpool (found by rosie99; in census 1851 and 1861)
Mary E born circa 1844 Manchester (census 1851) or Liverpool (census 1861)
Michael born circa 1846 Liverpool (census 1851 and 1861)
Anne 15 Jun 1847 Liverpool (found by rosie99, not in census)
[From Daniel and Mary:]
Thomas born 1850, Liverpool, in GRO (mmn Caton; in census 1851)
Daniel born 1852, Liverpool, in GRO (mmn Caton; in census 1861 and 1871)
Patrick born 1855, Liverpool, in GRO (mmn Caton; in census 1861 and 1871)
Ann born 1856, Liverpool, in GRO (mmn Caton; in census 1861 and 1871)
Margaret born 1859, Liverpool, in GRO (mmn Caton; in census 1861)
Elizabeth born 1860, Liverpool, in GRO (mmn Caton; in census 1861 and 1871).
Help in finding the first mother's maiden name (and explaining why the children are not in the GRO) is welcome.
Nicolaas