Four points seem most salient in discussing Swords & Wizardry Initiative mechanic:
(1) Initiative is d6 governed
(2) Initiative is decided by two opposing rolls (one for monster(s) vs one for PCs)
(3) Initiative is rerolled each new combat round
(4) Ties in initiative roll do not necessarily have to be resolved. Simultaneous action is possible.
Here are my initial thoughts on the suitability of this basic mechanic for our purposes.
First, I like the d6, which seems very old school and very s&w. But I fear it will have to go if we tinker with (2) as I am tempted to do. You see, I don't really like making a single PC roll being determinant of all PCs, since it still leaves unresolved the order of action of the PCs themselves. I understand that this could be determined via on-the-spot improvisation, but given (3) you'd be winging the ordering of PCs actions all the time. Why not allow individual PCs to roll for initiative each round and allow for some strange ordering patterns and associated challenges that will follow? (e.g., Blaine the Thief reached 0 hit points the previous round and Johann the Cleric decides he wants to try to make it over to the thief and cast a Cure wounds during the next round. This relatively straightforward action could be complicated by a poor initiative roll on the part of Johann...etc.)
One problem, though, with changing (2) and allowing individual PC initiative rolls (at least in any game s&w game we run in the near future) is that the number of PCs is likely to reach or exceed 8 at the minimum. 8 opposing d6 rolls is going to produce ties and time and time again. Consequently, I wonder if we adopt individual rolls, shouldn't we change (1) to some other die? Thus even a little tinkering in this way leads us further from the obvious "old school" trappings such as the ubiquity of the d6.
I'll offer some more thoughts in the comments as they come to me. I do find it very interesting that the LBB and Greyhawk supplement don't seem to have any initiative mechanic to speak of. Perhaps its the notorious lack of organization, but I don't seem to find anything on the subject.
What are the mechanics in the subsequent editions that are worth mentioning?