You can't do it with 1571, nor can you increase the number of rings before it kicks in. I would advise cancelling 1571 and spend the savings on a handset with proper answerphone.
I want to keep 1571, which is useful, but turn off the daft phone message telling me I have a voicemail. These calls come 10 minutes after a message has been left. BT Chat doesn't know how to do that.