24 hour TWOV on separate tickets

You have correctly concluded that your flight itinerary is eligible for 24 hour visa free transit, but you should have in mind that your entire transit time counted from your scheduled arrival in Shanghai until your scheduled departure from Shenzhen should be shorter than 24 hours!

Also, it is important to understand that it is not important whether you have separate tickets or fly on unrelated airlines because it is only the flight itinerary that really matters.

As far as your check-in in London is concerned, I am more that convinced that those working at check-in for China bound flight have at least some basic knowledge about 24 and 144 hour visa free transit schemes and if they are not certain, they can easily find an confirmation by taking a look at Timatic. Also, if you encounter a person not familiar with the matter, kindly ask for his/her superior to resolve the subject, but do not give up because you have full right to fly on such itinerary absolutely visa free.

Upon your landing in Shanghai the first thing 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. This application is free of any charge and a permit is necessary because your following flight will be a domestic one. When your permit is granted, you will proceed to the immigration and then to the gate for your flight to Shenzhen. Yes, you will also collect your luggage and perform new check-in for your new segment.

Upon landing in Shenzhen, you will cross exit immigration (exit China) and head to the gate for your flight to Los Angeles. Basically, this is it. Have a nice journey.