Sounds about right. I would do a close inspection of the queenless hive, most likely there are not just one but multiple queen cells by now. If not I would add a frame of fresh eggs. One queen cell is too few to bet on in my opinion. If there are many queen cells you have the option of splitting them into another hive, if you have plenty bees.
Once having done that leave them alone. The timeline is 21 days from egg being layed to emerging queen. It will be a month- mid August before there is a fertilized queen laying eggs.
Splitting now is a great way to drop the mite population. However, these hives need to be strong with many bees to build up enough before winter.