Do you mean your buildings insurance?? If so, I think it's tied to the property, not the mortgage, and as such I don't think you have to change it but you may need to amend your policy to get the new mortgage company named as a 'beneficiary' on the insurance.
If you are referring to buildings insurance you just need to inform your current insurer's who the new "interested party" is. Ie Jo Bloggs Bank
If your referring to Life Assurance and you want to increase the sum assured then there are certain circumstances where you can increase your current policy, or you may need to take a fresh policy for the full amount, or if premiums are going to increase too greatly it is possible to get a policy for the extra borrowing if you wish dependant on how much it is. For example my Life Assurance company have a minimum premium and for that I had to take a policy out for 100k.
Of course if you don't want to change any of that apart from your lender you just need to call them and tell them your new lender is X and they will put them down as an interested party.