London - Shenzhen - Phuket - 20 hour layover - can someone please help with correct transit visa?

Actually, you will not need any kind of visa because your flight itinerary or better to say its segment through China will be covered by 24 hour visa free transit and not a single type of visa is involved.

Now, you have to be aware that when traveling through one country, China included, you have a right to a transit that is visa free, but it also implies that you will not leave the transit area. However, Chinese authorities have implemented a temporary entry permit that is granted to those traveling with small children, disabled passengers and those having a very long layovers, especially during the nighttime. This permit, by its definition, is not a passenger's right, but more like a bonus that is offered on case by case basis. In other words, you should not contact the airline because there is nothing airline can do about this because this remains in absolute jurisdiction and discretionary decision of the Chinese immigration authorities.

Since you have 20 hours long layover, I would say that you will be granted a temporary entry permit almost certainly, but again, not with 100% certainty. There is nobody that could guarantee 100% certainty when we come to this option.

In any case, the first thing that you will do is to approach 24/144 hour visa free transit counter where you will apply for a temporary entry permit by presenting your flight tickets and your hotel booking. It is advisable to have all those details printed out. This application is free of any charge. After a short check, you will be granted your permit (almost certainly) and after that you will head to the immigration. When you have cleared the immigration you will proceed to the city to enjoy your free time on your own. Have a nice journey.