If the instructions require a 20A circuit then that is what you must do. If they recommend a 20A circuit but it doesn't actually require it based on total power, then a 15A is OK. Two keys:
Have you ever noticed that nearly all plug-in space heaters are rated at 1500W? Doesn't matter if they have a fan or not, ceramic, oil-filled, whatever. They are always 1500W! Why? Because 125V x 12A = 1500W. And a 15A circuit can only have 15A x 0.8 = 12A continuous usage. Heaters are assumed to be continuous usage because you might plug them in and walk away. Same thing with a bathroom heat fan. 1300W is actually on the low side, but that may be to allow for the exhaust fan and for a 100W incandescent light bulb, getting the total very close to 1500W.
Based on all of that, I am fairly certain that this device is going to be OK on a 15A circuit.
For those who might think "15A circuit breaker + 14 AWG wire, what's the possible problem - wouldn't the breaker trip if the device drew too much current?", it is not, unfortunately, quite that simple. If you have a 16A continuous load (max. for a 20A circuit) and you put that on a 15A circuit, the breaker might never trip because of the way breakers are designed. 30A and it will trip very quickly, 20A (33% over) and moderately quickly, but 16A is in the sweet spot where it might not trip for hours or never at all, but which would potentially be a major problem for the wire (which is why you are only supposed to have 12A continuous on 14 AWG wire). As with many other things, just because it seems to work doesn't mean it is safe!
In the end, manufacturer instructions and nameplate ratings are what really matter.
And finally, an update based on the specific model:
If it is this Broan NuTone 80CFM with 1300W heater, model # 765H80L, available at Home Depot:

then there is no problem, as the instructions say:
Provide a separate 15 AMP circuit. Use 14 GA. power cable of type which meets code. If providing a separate 20 AMP circuit, use 12 GA. power cable.
However, if it is this Broan NuTone 80CFM with 1300W heater, model # BHF80, also available at Home Depot:

then you must use a 20A circuit as the instructions say:
Provide a separate 20 AMP circuit. Use 12 GA. power cable of type which meets code.
This is a perfect example of why at DIY SE we ask for model numbers. Just the manufacturer (Broan-Nutone) and the basic specs (1300W heater, 80 CFM fan) are not enough!
Assuming that OP's problem is they have a BHF80, it is likely that they can swap it for a 765H80L and solve the electrical issue. Leaving a BHF80 in place with a 15A/14 AWG circuit is a code violation. A fire 10 years from now, for example due to accumulated dust or other problems, could be denied coverage based on an investigation showing invalid wiring.
All that being said, it is the responsibility of the electrician to match the circuit to the device, assuming that the electrician was either (a) doing the actual device installation or (b) was given full details and not just "Broan-Nutone 1300W 80CFM".