I'm taking a flight (Delta Airlines) from the United States, landing in OSL (Oslo, Norway), getting on a flight with a different airline (British Airways), which will take me to my final destination in the UK. I have no checked bags. I know I will have to go through customs in the UK, but will I have to go through customs and security in OSL to get to my other flight, or will I be able to stay in the terminal and switch gates?
Since you have no bags, no customs. Passports can be bypassed but it usually doesn't save any time.
All non-Schengen aircraft park at the far end of the international pier. That includes your flight and your onward BA flight.
The easiest way, if you have a U.S. or EU passport, is to go through passports into the Schengen zone, follow the corridor to transits, go through security again and then passport exit control (again) back into the non-Schengen pier.
Now at the first passport control there is a little door that takes you straight back into the departures security without any passport control. If you have a passport that is going to cause a lot of problems, you should use it. You ring the intercom and explain you are doing a non-Schengen to non-Schengen and eventually an agent comes along and opens the door. But if all the passport agents are busy processing passports it might take a while before they have the time to let you through. So for an EU passport holder it can be quicker to go through passports.
What you're talking about can be done in three minutes if you are at the front of the U.S. inbound aircraft with an EU passport. If you are at the back behind everyone else it might take twenty minutes.
You should obtain your BA boarding pass on the BA App on your phone to save time. Otherwise go to the BA gate ASAP and explain that you transferred and do not have a boarding pass.
Here's the instructions for OSL https://avinor.no/en/airport/oslo-airport/plan-your-trip/transfer/
Contact the border control for help to get to the non-Schengen transit area.