Often a broker would act as middle man and make initial contact with airlines to source availability. However, in the last minute case scenario, airlines tend to deal direct with other airlines. Under ACMI deals, the airline coming to the rescue will provide these ACMI items, in basic terms:
A = Aircraft
C = Crew
M = Maintenance
I = Insurance
They would fly all sectors on the client carriers flight numbers, handling would be with their approved handler, crew HOTAC would be booked and paid for by the client carrier, catering would be uplifted (as required) by the client carriers supplier, fuel would be uplifted to the clients account etc etc. All costs therefore go to the client and thus the rescuing airline is liable for very few costs except those associated with ACMI.
Payment can be tricky if operating on a fully inclusive basis and would depend a lot on who the rescuing airline is flying for. BA for example would be no problem for making payment (one hopes) but the deal would generally be under ACMI terms. However, if Fly-By-Night Airways requested a sub on a fully inclusive basis, payment would no doubt be requested up front by means of a wire transfer to a dedicated account. Unless payment was received and cleared, the aircraft would no doubt not depart. I've known aircraft to be sat waiting ready to be fuelled for hours while funds cleared. That then posed another problem, the crew were running out of hours to complete the sub. BUT, no payment means no deal.....
Arranging any sub is very straight forward generally. Ops departments receive the nod from commercial to sub out a flight and once a suitable deal has been reached, it's then left to each airlines' ops departments to fine tune the necessary arrangements either via e-mail, or SITA messages and of course via phone. Good ops teams work swiftly and efficiently to get subs under way so as to keep delays down to a minimum.
I remember asking easyJet to delay one of their flights so I could re-accommodate some stranded passengers of mine who had returned to base with a tech aircraft. They agreed which just went to prove that locos can actually work together. We paid an agreed seat rate per pax and signed the deal, all within 30 minutes. Whether or not that would happen these days remains to be seen......
Hope this helps.