Thomas is 3 months old on the 1841 census. He and his mother (who was 25 at the time) are living with her parents, John and Sarah Beacock.
In 1851 Sarah is living with Thomas. Sarah is now Sarah Gilliat after marrying John Gilliat in 1845 (4 years after Thomas was born). He died in 1849. I have thought that John could be the father, but I could never be sure. The marriage does seem quite quick, so it is possible. Sarah's occupation is Annuitant which I suppose could mean that John could have left some money for Sarah and Thomas. Maybe an act of goodwill, or because Thomas was really his son? I don't know.
A few months after the 1851 census was taken, Sarah married Gilbert Burton Handson.
In In the 1861 Census, Thomas is described as being the "Son-in-Law" of Gilbert Burton Handson, Thomas is Gilbert's stepson. I believe that Thomas had a good relationship with Gilbert as Thomas went on to be a butcher like Gilbert, which suggests that he worked with him. I also think this because, on Thomas' marriage certificate, he lists his father as Gilbert Beacock. There is no such person, at least at that time, so that doesn't match up. All I can think of is that Thomas wanted to cover up the fact he was illegitimate. He did move away to Hull, from Lincolnshire before he got married. This could have been for a new life.
There's a lot of ideas but I just can't pinpoint anything. I have also done the AncestryDNA kit. I suppose I could look further into John Gilliat's family and could get a link to him. If so, I would know that he was the father.