144 hour free visa for UK passport holder

Unfortunately, this is not possible. When a visitor uses 144 hour visa free transit scheme, she/he is limited to just one Chinese transit area. Since Shanghai and Shenzhen are not in a common transit area, your itinerary would have required a visa obtained in advance.

In reality, you can alter/adapt your itinerary to look like this:

London - Shanghai (144 hour visa free transit) - Hong Kong

From Hong Kong you can enter Shenzhen overland and apply for a 5 days Shenzhen visa right at the border crossing (small drawback for UK citizens is the fact that Shenzhen visa on arrival cost is 834 CNY dur to reciprocity). One additional convenient detail is that you can depart from Hong Kong airport Skypier directly by ferry to Shenzhen Shekou port where you can obtain a Shenzhen visa on arrival. However, have in mind the opening hours of Shenzhen Shekou visa office (9:00 - 13:00 & 14:30 - 16:30).

After you have accomplished everything you have planned in Shenzhen, you can return to Hong Kong Skypier by ferry at any time and from there take a flight back to Shanghai and from Shanghai to London. Maybe this could be your solution for your itinerary.

In the end, I encourage you to use correct terminology because it is 144 hour visa free transit and not 144 hour free visa... There is no visa at all... OK, that one for Shenzhen is a visa, but it is not visa free transit, but a visit.