We can see that various later editions tried to deal with this issue by instituting various solutions such as tracking initiative by Dexterity rating (Holmes) or by rolling a d6 (AD&D, Moldvay). Both of these approaches can be extrapolated from Men & Magic.
However--the whole point of the referee is to adjudicate these kinds of situations! So with that in mind:
First, the referee looks at situational factors like surprise. Does it seem likely that any of the participants are surprised? Is there a chance that the PCs may be alerted in some way? How do we determine this? The referee adjudicates the situation FAIRLY based on situational factors. That is, I think, the default system implied by the OD&D rulebooks.
Second, assuming that combat has been joined--what happens next? Who goes first? Well how fast and skilled are the PCs? (this is where Dex becomes a factor as it indicates in Men & Magic) Are there other factors that need to be accounted for? If the PCs are heavily armored Dwarves using heavy axes fighting a group of Orcs wearing leather armor and armed with lighter short swords and crossbows--might the Orcs gain initiative? Or what if one the players is so quickly decisive that she immediately declares that her PC is charging the Orcs and yelling the ancient battle cry of the Dwarves? Might this PC gain initiative before the Orcs even if the other Dwarves do not?
Other factors such as weapon reach etc. can also be factored in through adjudication.
If it looks like it’s a too close to call--you can always fall back on the d6 as a "tie-breaker" as Holmes suggests.
Other factors such as additional strikes (ripostes etc.) can be adjudicated as well. In this regard I'd use Chainmail as a guideline as to situational factors that might play into this.
The referee or the players makes attack rolls as usual with the DM adjudicating the effectiveness of hits etc. Use d6 or variable damage here as you will. One of the main jobs of the referee here is to make combat vivid and exciting. Don't just drone on about hit points lost--give us some real description of what's happening! D&D combat is very abstract, and it is up to the referee here to make it sing. Season with whatever level of gore you and your players are comfortable with. Bad things happen when people hit each other with sharp pieces of metal!
This approach really comes out of OD&D's wargame roots but it also veers close to what's today called "freeform." The referee has as many tools as they want here. Lots of folks might feel like they're flying without the net of the rules here--to that I say: Welcome to OD&D!
Lots of folks have obviously seen the advantages to this approach which was lost to one degree or another in all later iterations of the game. Indeed, so many people complained about this to TSR that they felt they needed a more solid rules based edition that didn’t leave so much to the referee. Thus Advanced Dungeons and Dragons was born. I love AD&D but there’s something very freeing in all the open spaces of OD&D.