Dealing with 15-minute transaction time limits
It sounds like you are using two different hosted wallet providers.  What you should do is download your own wallet application that you run yourself, and use it as an intermediary between your two hosted providers.  This can be a wallet that runs on a desktop, laptop, or your phone, but doesn't have to be a full node.  You can find a wallet here:  https://bitcoin.org/en/choose-your-wallet

Once you have that, withdraw your bitcoin from the first service to an address from your own wallet.  This may take the usual several hours.  Once you see that the transaction has a confirmation on it, you will be able to then send it to the second hosted wallet easily within the 15 minute timeout.