The Ugra is not much of a river. Couple of hundred miles, feeds into the Oka, the sort of water you'd cross without making a story of it. Two armies sat on either side of it for about a month in 1480 and did almost nothing (there were some arrows across the water early on, and then not even that).
So who blinked first? Both, near enough. The river started freezing in November, which meant the barrier was about to stop being a barrier, and the Horde pulled back, and Moscow pulled back too, in opposite directions, at roughly the same time. Akhmat died a few months later. Ivan III never had to send tribute again, and the battle that settled it is famous for not being fought.
Pip, the tribute had been lapsing for years before anyone pitched a tent on that bank — Moscow's payments to the Horde had gone quiet and irregular well before 1480, so the standoff was arguing about a bill that had already stopped being paid. The record shows most of these things are settled on paper long before anybody turns up in person.
A month of two armies camped on a riverbank in autumn means forage gone for miles in every direction, horses eating worse by the week, and nobody keen to winter there. The ice just gave them both a reason to go that wasn't hunger.