I base my following remarks partly on the contents of the two podcasts I mentioned earlier. In it information is disclosed from Mary's surviving family members (her siblings, who were all younger than her); it is worth emphasising that their recollections are a mixture of their own first-hand experiences and hearsay from their parents and other family members.
The first thing that I will address is Mary's absence from work for the fortnight preceding her disappearance. I suggested in an earlier post that she may have taken this as leave. It seems almost certain that no one from her employer (Tate and Lyle) which was a somewhat paternalistic employer, made enquiries about her absence which suggests to me that they were aware of her reasons for staying away from work. Had she just absented herself without explanation then she would be taking a huge risk of someone making enquiries at her home during working hours and revealing the deception to her family. If she wasn't on annual leave then she would have presumably phoned in, or even written a letter, (the postal service was much better then) to her employer to explain that her absence was due to something like illness.
To explain her disappearance it has been suggested that she eloped with Tom, her putative fiancé; this makes no sense. If they had eloped then Tom would have disappeared at the same time as Mary. We are told, however, that following her disappearance Tom helped her father look for her, so elopement with Tom in the normally accepted meaning of the word didn't happen. Of course, Tom may have known of her whereabouts and his actions may therefore have been simply misdirection.
Mary and the Flanagans were led to believe that Tom was staying in digs but the woman who supposedly was his landlady was in reality his mother. When the true relationship was discovered shortly before Mary's disappearance it was said that there was a serious row between Mary and her father about this.
So we have two apparently motiveless deceptions - Mary's pretence of attending work and Tom's domestic arrangements.
Here is a possible scenario: Mary had been planning to leave home because she didn't want to marry Tom (he was her father's friend before he was her fiancé) and she may have been coerced into her engagement by her parents. She had secretly got another job early in December and was going there every day rather than to Tate and Lyle. Perhaps the row about Tom had nothing to do with his mother but came about because Mary felt at that stage she could tell them about her attitude to marrying Tom.
It is not clear what she took with her when she left home for the last time: if she had not been working at Tate and Lyle for a fortnight then she would have missed two paydays (I think it highly likely that she was paid weekly in cash) but my guess is that she had adequate funds from her new employer to pay for digs. Perhaps she spent New Year's Eve afternoon shopping for clothes etc prior to moving in and went back to work, with her new employer, on New Year's Day.