If they are telling you that they can check your bag through, that is good news; however it conflicts with info on the BA website
https://www.britishairways.com/en-us/information/baggage-essentials/checked-baggage-allowances
”If you’ve booked separate flights with different booking references to get to your final destination each flight is considered a separate journey. This applies even if both of your flights are with British Airways. You’ll need to check your bags in for each separate flight. Make sure to leave enough time between each flight to do this and get to your departure gate for the next leg of your journey. Simply follow these steps:
Check in your bags for your first flight
Collect your bags when you land - you may need to go through customs
Go to departures and check your bags in again for your next flight
Collect your bags at your final destination
Please be aware that on journeys with separate tickets and different booking references, your bags will not be transferred on to your next flight as it is considered a separate journey.
In case of delay, cancellation or other disruption to your initial flight, when you hold a separate booking on a different reference for your next flight, airlines are limited in the assistance they can give you for your onward journey/subsequent flight, just as they have no obligation to automatically check your baggage through to your next flight as each booking is considered a separate contract.”
Many years ago, I was able to check bags through on separate tickets on BA, but they subsequently changed there policy. Many of the airlines changed their policy after 9/11. At Heathrow, you can get through the egates relatively quickly and then will just have to wait for luggage. Four hours should be enough time if your flight arrives on time. Hopefully, the info provided to you verbally is correct and this will be a non-issue, but I’d plan for the worst case.