MoneyM,
As long as your new mortgage is for the same amount and term as the old mortgage, I think you can leave your mortgage protection policy in place and have it assigned to your new lender. (I'm open to correction on this!)
However, this is not the case if you got your policy through your lender's group scheme. If your policy is part of the lender's group policy, it is usually cancelled and you would have to go and get new insurance, medicals etc. as you outlined in your post.
However, if you are not tied to a group policy and not topping up your mortgage, or changing the term, you can hold onto your policy.