I'm pretty sure that when taxing your vehicle online, there is a disclaimer stating that you should not use the online serivce if you have moved recently. It also says this in red writing:
"If you have recently purchased this vehicle and you have not received your full Registration Certificate (log book) from DVLA (the green slip alone is not enough) you should NOT use this service.
If you do use this service without having received your full Registration Certificate (log book) from DVLA then your new tax disc may be sent to the previous keeper of the vehicle.