She did, when the registrar came to the maternity hospital. If you are still doubtful get a copy of the local birth register entry and compare it with previous examples of Maria's signature.I have a doubt about the signing on 29th August by M A Grounsell if Maria was still in hospital - who signed?
Many burials from that era are difficult to find, and it's not unusual for people to be missing from the census, or simply hard to find because they're not with their parents. Also consider the possibility that they emigrated, or were with their mother, who had also remarried.I have checked Lunacy Patients at Ancestry also the LMA and TNA and anything else I could locate.
As an aside James's daughters Emily b1850 (grousell) and Ellen Ann b1851 (gronsell) have not been able to locate their deaths or burials, neither was around for the 1861 census it must be something about the name Grounsell!