Travelling from Hong Kong to Shenzhen on a British passport for an overnight stay

Yes, you need a visa that is called 5 days Shenzhen visa on arrival and costs 834 CNY for UK citizens, based on reciprocity between UK and China.

You cannot take a direct, high speed train, if you were thinking about that one. You can take a train until Lo Wu (Luohu) border crossing and apply for this VOA at the border crossing.

Anyway, the best way to reach Shenzhen is to take the first morning ferry from Hong Kong to Shekou in Shenzhen and to apply for a VOA exactly there because it is least congested visa office of all.