Quotient farthest mouths/nearest sources
Tbb is correct. The 'Congo and Nile' is by far the winner. Their mouths are 4600 km apart, whereas if you trek into the Rwenzori mountains, Western Uganda, there is s small highland swamp in a saddle point on the Congo-Uganda watershed. On either side there are several points where you can claim to be 'at the source', where you can stand with one foot in the Congo basin and the other foot in the Nile basin. The Indus and Brahmaputra are less than half this distance apart.

As for the other extreme, it's impossible to say. There are several rivers which end at more or less the same place. One then has to argue, 'Is one river a tributary of the other, or not?'