I don't think that there is any reason to suspect that Vortigern's coronation would have involved elaborate ceremonies or oaths. The normal procedure, for Germanic kings at least, was for the new king to be raised up on the shields of his soldiers and carried round the tribe in procession several times. He would then be presented with a diadem and a royal spear as a symbol of his authority. Incidentally, this is a similar ceremony to the one performed for Roman emperors. After the coming of Christianity it became common for kings to the nearby monastery or cathedral to be crowned and be crowned by the abbot or bishop. A large feast would then follow. It might be that a prospective king would tap into the memory of fallen Rome by making their court/capital in or around a Roman centre of power, as Edwin of Northumbria did at York. The imposing presence of these large structures, now outside the capabilities of the people might well help to shore up the authority of the new king.

The Conversion of Europe: From Paganism to Christianity, 371-1386 AD, Fletcher