The normal procedue if its still the same from the time I worked in credit control
the normal procedure is a reminder is sent then after a time outgoing calls barred (with the exception of 3 digit service numbers freephone and emergency numbers, then tempory loss of service (complete barr) then a few more letters before a termination notice which leeds to the final bill, to get to that stage is a looong way down the line a lot more than 3 weeks.
Typicaly you can call to arrange a postponed payment plan and this is easly done and typicaly before or about the time of outgoing calls barred and with an prior arranged agreement all following actions can be suspended. Sorry to say but I suspect there's a lot more to this then you were told
I remember one chap who insisted as he offered to pay cash direct to a bt employee it was bt's problem the payment couldn't be taken that way so as he offered to pay he couldn;t be cut off. Wrong, thats when we worked out at the time there was 28 different means of paying the bills