Sort of worked part of it out. Tom Bapty SHARP was born in Leeds in 1869. Was in Durham in 1891. Married Lillian Josephine TOOKEY in Dec qtr 1899 Newcastle T 10b 259 and was with her and her mother in 1901 (RG13-4170 Folio 81) with child Lilian B SHARP aged 10 months.
The marriage certificate of Tom and Lilliam might tell you whether he was widow or single.
No sign of marriage of Theresa to Tom B SHARP though they may have been married in Scotland. No reason why they should though as she was over 23. If Tom and Theresa were married then where is her death? there is a death of a Theresa SHARP in 1898 but age was 0 - mabye it should read 30.
Possible that the marriage in FreeBMD in Dec 1891 Aukland 10a 330 to either James SENNET or Harland READMAN was your Theresa. Then either her husband died or they separated. Would be great if there was then a marriage of Teresa SENNET to a Tom SHARP but I cannot find one.
Loads of records of births of Dorothy SHARP in the 1890s but which one is which? The IGI has a baptism of Dorothy Annie SHARP to Tom SHARP and Theresa 23 OCT 1892 St John, Manchester - But it is a funny Batch number and so is suspect. However the birth is registered as can be seen in FReeBMD and so her certificate can be obtained and may confirm whether Tom and Theresa were married or not.
I think it would be worth paying up for the certificates for the marriage of Tom in 1899 and Theresa in 1891.

In case FreeBMD has just not indexed the marriage of Theresa and Tom you could try searching the indexes manually for the couple of years from March 1891 if you have not already done so. Could be one of those rare occasions where the record was never sent to London in which case the local registrar might have a copy but which registration district?

Did you find Dorothy and her siblings in 1901 - I could not. Who they were with may help undrestand what was going on.

Time for bed.
Bryan