It depends largely and where you live and how much you need (or want) the wood. There are several reasons to wait till winter, but they may not be an issue for you. One is insects. If you are cutting at this time of year right as the tree is starting to produce fruit, the wood will be sap heavy and will attract insects looking for the glucose in the sap wood. The heart wood will of course be insect resistant in mulberry, but the sapwood will go through all hell. If you wanted a heart-wood sap wood bow, that would be a problem. The second is checking. If the wood is super wet, it will likely check heavily as it dries. Even if you seal it. I am not sure what would happen if you immediately reduced it to near dimensions. Perhaps nothing. I usually split it into staves. I primarily cut mulberry where I live and stopped cutting it in the summer after it seemed spring and summer cut wood was twisting more as it dried. I cannot rule out that this was simply grain distortions and nothing more, but I just choose to wait now. A lot easier in the winter for me too since the hornets and snakes are dormant and the temperature is below 100! Either way, it is an endearing wood and excellent fruit. You can basically cut it to a stump and it will shoot back the next year too. Sustainable.
