If I’m understanding your post right, after switching service to BT Mobile you have had a constant red dot on your phone app. I can certainly understand how that red dot could become a nuisance after being there for awhile. You’re in the right place for a resolution.
That red dot is usually indicative of a problem with your phone connecting to your voicemail, rather than a notification that you have new voicemail messages in your inbox. If voicemail was working properly, you should have a number badge on your Phone app telling you how many voicemails you have. You have already done some great steps to resolve this. Next, I recommend resetting your voicemail password, which can help trigger voicemail to remove the red dot. To do this, go to Settings > Phone > Change Voicemail Passcode. If there is a passcode already set up, you may be prompted to enter it in first. If you don't remember what the passcode is, please follow the steps in If you forget the voicemail passcode on your iPhone .
This should clear that red dot and make it so there is a number badge on your Phone app when there is an unread voicemail.
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Take care.