Godwin's diary Vol VII f 46r Sidney Montague for India, 1788 - refuses 5 guineas of BM, & takes 5s - about 21
See Gentleman Radical by C & D K Bewley, which cites Mrs Thelwall's Life of John Thelwall p77 and Samuel Rogers' Recollections p145. Sidney Montague was said to be Horne Tooke's illegitimate son, but I can find very little evidence for it in the sources that the Bewleys cite, they presumably had other sources that they didn't cite, particularly for the events involving Basil Montagu, who must be the B M above. Godwin's note above may be for all I know one of the chief sources.
Harriet Montagu was a courtesan portrayed in John Raphael Smith's series of engravings of six named ladies in 1777 and was also identified as one of the ladies in John Raphael Smith's Promenade at Carlisle House (1781) in Henry Bromley's Catalogue of Engraved British Portraits. The will PCC 1806 of Harriet Montague Decaix of Amiens, France mentions her annuity from Lord Courtenay. Sidney Montague would have been born about 1767 but it is possible she was his mother. He may also have been the son of a sister or aunt of Basil Montagu, the spelling with an e or not being unlikely to signify much. There is no record in Alum Cantab of any student of that name, or called Horne or Tooke. Tooke is supposed to have offered the rooms at St John's College Cambridge (Tooke's own college) that his son had occupied to Thelwall. His son was maybe expelled for bad behaviour before even being admitted or matriculated or was struck off the record. One corroboration of the story of Sidney Montague being accessory to a robbery can be found in the World 16.4.1787 but the online newspapers contain no further reference to this case