Given the facts as you presented them, the hive will die unless YOU introduce a new queen. A laying worker will not mate with a drone - she will only lay drone eggs. Two things have to happen in order for the hive to requeen itself. First, there has to be the general "recognition" (by the bees) the colony is without a queen and secondly, there has to be a female (fertilized) egg or larvae of the proper age to feed royal jelly to in order to develop a queen.
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Given the strength (population) of the colony at this time of year, it is possible your queen has just not begun to lay yet. There would be doubt in my mind if your hive is really queenless.

A hive makes an emergency queen from female eggs laid in the past two days. The queen is out in 21 days and flies out to mate a week or so later, so in a month, you could be queen right.

if

you have fresh female eggs

if

weather is good for mating, sunny 70s or better

if

there are enough drones to mate her well

if

she finds her way back to the hive after mating without getting injured or eaten.

Now that I have ventured a split with my stronger colony it prompts me to ponder....

What if the only drones around are from my other two colonies. Is this tantamount to inbreeding? Will my bees end up 'funny in the head' if done repeatedly?
Also, why only one mating flight anyways?

Since I understand that drones are in and out of eachothers colonies I suspect they would spot a virgin queen cell on the rise and be willing to assist. The intitally weaker colony seems to be roaring a bit louder lately.

Life is good.

I believe an emergency queen can be "created" from a larva as old as three days. ALL larvae are fed royal jelly the first two days of their existence. Only from the third day on are future queens continually fed royal jelly. Counting from day three, the larva will be capped in in five days, and the new queen will emerge eight days later. It takes 16 days from the day the egg is layed to a new queen emerging.

After emerging, the virgin queen should mate within seven days, and start laying eggs about a week after that.

And inbreeding with a son is avoided, as a signficant portion of eggs laid will not be viable. It doesn't take generations of inbreeding (son to mother) it happens immediately as the son is a clone of the mother.