I see from the 1841 census that a 10-year-old William Lyall, born in Angus, is listed in the Royal Orphan Institution in Dundee.
http://www.freecen.org.uk/cgi/search.plAlso from the Dundee Howff gravestones web site
http://www.dundee-howff.info/ that William Lyall, weaver, was aged 60 when he died in 1838. If his age is accurate, this means that he was born in 1777/1778, and that he was over 50 when William Lyall Jr was born.
I also note that in the same grave there are other people with different surnames and no obvious connection to William. Now, I may be adding two and two and making five (or even seven) but it occurs to me to wonder if William Sr had fallen on hard times, and been buried at the expense of the parish? If so, there could be something in the Kirk Session records (1838 is too early for Poor Law records). Start with
https://www.dundeecity.gov.uk/archive/Also, if William Jr was indeed an orphan, there is an outside chance that he was still at school in 1845, when the Poor Law came into force and the Parochial Boards were set up. If so, there might be some record of him there. Again, start with
https://www.dundeecity.gov.uk/archive/I don't think there is any need to check that death in Panbride in 1858, by the way. The SP index shows that her other surname was Fitchie. Barbara Patullo married John Fitchie in Tealing in 1812. They had four recorded children, George, Barbara, John and Elizabeth, and in 1841 they were living at Monikie with Elizabeth (listed as Betsy) and three unrecorded children, Margaret, James and Helen. So there is no possibility that she is your Barbara Patullo.