Wherever these folk got to I do not think it was "Hamburg to Leith via Glasgow"!!. Leith was one of the feeder ports on the Indirect Route, Hamburg - Leith - train to Glasgow, c.77km, 1 hour. Other way round? - not unless they were in a "Round Britain" race!!
I think the lack of information about ships to South America is due to the fact that it was unimportant in the emigrant story and very few people went there from UK at any time. Brazilian Railways, and others I think, were built by the British and, I understand that the Brazilian textile trade was mostly British enterprise, so there was a fair amount of business but little emigration. The main shipping line for South America was the Royal Mail Lines, started as the Royal Mail Steam Packet I think, changed name c. 1932 when the ships were definitely sailing from Southampton but I think it was founded c. 1832 in Liverpool. Would have to check up on the details, but either Liverpool Merseyside Museum or Southampton Local Studies are likely to have any records that exist - with the caveat that So'ton would only have records for after the move. Have had a casual look and cannot see anything. Shipsnostalgia only brings up contacts from people who worked for RML in its final days, no historical stuff.