Is there a SAFE place to store a cellphone, camera, etc, while visiting the Old Bailey?
I don't see any reason you can't have them with you.
https://www.cityoflondonguides.com/tours/the-old-bailey
You don't need to store phones or cameras you're just simply not allowed to take photos in any court in the UK. Everyone will have their phones with them in court from the public to lawyers and even judges, just turn them to silent and all will be fine.
https://www.cityoflondon.gov.uk/assets/about-us/law-historic-governance/prohibited-items.pdf
Mobile phones (and cameras) are not allowed in the Criminal Court.
There will be a difference of approach depending on whether you are attending a court case, or if you are on one of the weekend tours. So it would be helpful for the forum if you could confirm which you are doing.
If you are attending a court case and the prohibited list is right then look for a Stasher type luggage storage location- two of the very close by Premier Inns do Stasher luggage storage. Or even better just leave it in your hotel room.