Do tax refunds from no TFN have interest?

I swear I gave my employer my TFN but they don't have it on their system and I never got a confirmation so I can't prove I did. In any case, as far as the ATO is concerned, they got half of my pay and no TFN associated with it.

I'm bummed because it could have paid rent but I've been told nothing can be done until tax return time.

My employer now has my TFN but says payroll can only fix it in January, which means I will have a couple of months' overpayment. Is this considered "early payment" of income tax by the ATO so I can get it back with interest?