Tom McGregor married in 1917 Wandsworth to Bertha Ellen Lenton.
You need to see the marriage certificate for his fathers name and occupation.
Do You have that record?
Since you havent answered this question, we assume you need to acquire the marriage record for the information it should provide about his father.
Once you have it, it might be possible to help with additional information.
Some general thoughts and observances:-
Trying to find his brother John McGregor through the army will be tough. It is such a common name in Scotland, and John may have moved from one regiment to another, which was fairly typical. The majority of British WW1 records were destroyed in WW2 bombing.
From Thomas McGregors WW1 military papers we know that he was a labourer, and probably had limited ability to read or write. This is worth remembering when looking for passenger lists or census records. His family background would probably be the same.
I would not take the age on Thomas military papers as necessarily accurate. In my experience, ages were often altered. Younger men put their age up, and older men put their age down in order to meet criteria. The photo of Thomas in his WW1 uniform (attached to the public tree) could quite easily be a man in his 30s, not his 20s.
His WW1 papers also include several enquiries about his wellbeing from Mrs. A. M Sheridan, and her daughter. It is obvious they were dear friends, but I could not see a family connection. James Sheridan (1869 -1925) married Annie May Day (1869-1958) in 1892 in QLD. James Sheridan was a labourer and appears to have worked away from home for the railway, while his wife and 2 children were based in Rockhampton from about 1912-1922. It looks as if Thomas McGregor was a close friend of the Sheridan family and possibly a co-worker of James on the railway. The construction of the railway in central Queensland at this time employed a large number of labourers over many years.
As you are aware, McGregor is a very common name in Scotland. However it is possible that your Thomas was born with a different surname, either illegitimate or his mother remarried and the surname changed. McGregor may not necessarily have been the name that he came to Australia under.