|If you go to http://www.hounslow.gov.uk/index/transport_and_streets/parking/parking_permits.htm, and scroll down to "Change of vehicle", there's a link to the form.|
It's the same form as for a courtesy car (when your car is temporarily off the road). You get a reference number which you have to display on the new/temporary vehicle with your existing permit. I assume that if you select the "New vehicle" option on the form, they will send you the new permit in due course. Meanwhile the reference number plus old permit will be valid for 4 weeks.