When is it that you are flying? If you are within 48 hours of flying then you need to be on the case. If you are more than that, just leave it.

Online bookings are ticketed automatically, typically within an hour, sometimes a few hours if there is a non BA flight or some other complication.

Telephone booking often need manual ticketing. This can take months, and is rarely a problem. They are ticketed in sequence of departure date priority, and may even be unticketed a week before flying even if it was booked months ago.

I have - well let's say close to 100 reservations with BA at the moment. I'm not counting, but I know I have over a dozen unticketed too. It's not worth panicking about. The very small risk of being denied travel does exist, and yes it is worth taking action at some point. Unticketed bookings do very, very occasionally go down a blackhole.

If you search here you will find other Customer Service centres open at various hours. Maybe call Newcastle or Hong Kong at quiet periods, when there aren't storms on, and get it sorted then. If you are within 48 hours of travel you are able to ask the agent to wait and confirm ticketing.

The internet, including Flyertalk, are full of anecdotes, some of them a bit scary. That's not the same as data.