A quick look at the manual (here:
https://myfile.org/A0K8N6b5ba/TG7321E_PNQX1034YA.pdf ) confirms that your Panasonic phone only uses tone dialling.
The fact that the BT exchange can 'hear' the tones, but other systems can't, suggests to me that the 'volume' on your line is too low, with BT's highly-sensitive equipment able to 'hear' the tones but other equipment not able to do so. However, as you're using a cordless phone, it could be that the signal strength from the handset is inadequate for all systems to pick up on.
Try borrowing a few more phones, ensuring that they're NOT cordless. If the problem persists then it must be due to inadequate 'gain' ( = 'volume) on your line for outgoing calls, which you'll need a BT OpenReach engineer to fix. If the problem disappears then either the handset of your cordless phone isn't sending a strong enough signal or there's some nearby equipment (such as a printer) which is producing interference that's partially blocking the signal. [Many electronic devices which appear to be 'off' are actually only on 'standby' and emitting radio interference. Simply place a portable radio next to an 'off' printer to see what I mean!]