I would expect it to be done within a day or two... possibly a day or two more if it happened on a weekend. How long has it been?
If you think the landlord is going too slow, contact the landlord and ask if you can bring someone in to do the repair and bill the landlord.
Repairs should not take that long, although in the middle of a cold spell everyone is suddenly calling the repair guys and they are typically swamped with business.
Requiring an entire new unit is likely to take longer, depending on whether the unit is in stock, how long it takes to be delivered, etc..