If part-exchanging then any decent garage should sort it all out for you.
The only problem is timing the insurance - a reg plate can only be associated with one car at a time, so your insurer will not allow any overlap, even by a few minutes. So, be careful that if there is any delay in your new car being ready, then you will probably be out of insurance on your old car as you will have arranged in advance ( in order to get road tax ) for the new insurance to kick in.
Apart from that, put your feet up and let the garage take the hassle
Gus