All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
What is meant by conventional mortgage in the west is a usurious loan which the borrower receives to purchase a property, and the loan is "secured" by the property ownership. If this is what you mean and you have taken such a usurious loan, then you have committed a sin and it is incumbent on you to repent from that sin. If by repaying the loan before the due time the interest will be waived and you can avoid paying the interest, you should hurry to repay it. However, if you would still be liable to pay the interest, the early repayment of the loan before its due time actually works to the advantage of the usurious bank, because it will have taken back the capital early and the interest as well; in this case, you should not pay off the loan before the due date. There is no harm in transferring the loan to an Islamic bank as long as it does not involve engaging in any prohibited transaction.
We must warn you against making light of debt, even if the loan is taken from an Islamic bank, because the revealed texts generally discourage and warn against taking unnecessary loans, and urge Muslims to avoid it as much as possible except in case of a legitimate need. It was narrated that the Prophet, sallallaahu ‘alayhi wa sallam, abstained from offering the funeral prayer for someone who died in debt. Ibn Al-ʻArabi is quoted in Tuhfat Al-Ahwathi as saying, “That he abstained from praying for someone who died leaving debt was to warn against heedlessly falling into debt so that people's money does not go wasted.”
Allaah Knows best.