Newton Abbot Beekeepers

It has been a very strange season, but I have only split my colonies once. I generally take out the queens with 3/4 frames into nucs when the intention to swarm is clear and then leave the main colony for a week to produce their emergency Queen cells. The following week all the emergency Queen cells are knocked down and a frame of eggs from whichever Queen I want to breed from is put in. The next week I select the best Queen cell produced from my breeding Queens eggs and knock down all the rest. Then I can leave them for 4 weeks for the new Queen to hatch/fly/mate in the knowledge that they don't have any eggs to produce more Queens to swarm. This way I get past the main swarming season and still have a full colony to produce honey when the blackberries come out in July.

Ah, the 64 million dollar question!!
The aim is always to improve your colonies by making them stronger, healthier and pleasant to work with. So first you need to take a long hard look at both the old and the new Queens. 
- If either colony is aggressive, then don't keep that Queen. Aggressive behaviour is almost always directly related to the Queen.
- If the old Queen is into her third season, then her laying rate and pheromones will be reducing. This will lead to the colony replacing her if you don't and if this happens late in the season, there is a risk that a new Queen may not be properly mated and may fail in the Spring.
- If either Queen is not laying well, or has a patchy brood pattern, then it is possible that she did not mate properly, or is failing.
In any of these cases the offending Queen is a threat to the colonies survival and should not be kept. If this is the case then the two colonies can be reunited after the Queen has been removed.
If however they are both good healthy Queens and colonies and you don't want to have an additional colony or kill the Queen, then either;
- decide which Queen you wish to keep and advertise the second for sale, the purchaser to provide the equipment for the bees to be housed in  (Never sell bees if you are uncertain of their health or temperament.).
Or - keep the old Queen in the nucleus hive, keep the colony small by removing eggs/sealed brood and giving them to the new Queen's colony. This will increase the size of the new colony more rapidly so that you can take better advantage of the Brambles/clover flowering. You can then keep the old Queen in the nucleus hive for the winter. This gives you extra insurance in case your new Queen fails in the spring or dies over the winter, a time when colonies are at the greatest risk of failing, with no new Queens available. In the spring, sell/give away the old Queen if she is not reqired and is still healthy. There are always other beekeepers whose Queens become drone layers or die over the winter and are unable to produce a new Queen so early in the season.