Hello Peter, thank you for the info.
This symptom can indicate either a problem with the line (phone company equipment or the wiring in your home) OR a hardware failure in the base station (since the issue happens with all handsets). We most often see this type of damage to the base unit as the result of a power surge, either through the phone or power lines.
It would be very helpful to test another phone, if possible. Perhaps you can buy a cheap corded phone locally, or borrow one from a friend. If the issue happens on another phone as well, then it is a line issue, and BT would need to send an engineer out to check their equipment. If the other phone works, then we know it is a panasonic problem.
For a problem with the base station, the one thing you may attempt yourself is to fully reset the system:
1) unplug both power and phone lines from the main base.
2) remove batteries from all handsets
3) wait at least 15 minutes, then restore all
4) place a handset on the main base for a minute, as if to charge
5) check for functionality.
If the problem persists after this, it tends to confirm a hardware failure which would require repairs to correct. Unfortunately once the 1 year warranty is up, repairs are not practical, and the base alone cannot be replaced. The most economical option in this case is to replace the entire system.
At that point, what I would suggest is to buy another system of the same model
Panasonic KX-TG2513ET DECT Trio (click for link)
as this would at least allow you to retain some value from your existing system. you'd junk the base station of the existing system, and then pair your existing handsets to the new system, expanding your total handsets to 6 in the process (or giving you some spares, in case some are ever damaged or stop working).