Question: I live in an old apartment with radiators for heat, The gas boiler is down two levels in the basement. oddly enough there are no cold air returns...is this correct? Our carbon monoxide alarm continuously goes off when the heat is turned up, we cook on our ELECTRIC STOVE. If we forget to turn the heat down at night, we all wake up with headaches and nausea. What is your guess as to what is wrong.

Answer: 
If your carbon monoxide detector is going off, you have a carbon monoxide problem. Contact your utility or local emergency services for a more thorough test. The symptoms you describe are consistent with early stages of carbon monoxide poisoning.

If your radiators heat with hot water or steam, or another system other than forced air, you will not have return ducts. That is normal. Carbon monoxide inside homes typically originates from a gas-burning appliance directly -- so in your case it would be gases coming up from a heater or boiler (probably in the basement). Alternatively, it could be that the exhaust duct / chimney is leaking into your apartment, rather than completely exhausting outdoors.

Get this looked at right away by professionals. BE VERY CAREFUL and do not venture downstairs; if the CO problem is bad in your apartment, it might be much worse in the immediate surroundings of the heater. Be safe!
As your CO detector is going off: 

1. Get you and your family into the fresh air
2. Call 911

The fire department will come and check everything and figure out where it's coming from. They have an expensive carbon monoxide meter and they will check if there is a CO leak. If they find something they will call the gas company to shut off the gas until the problem is fixed by professional.

If your CO detector goes off, assuming you have the cheap one from Home Depot, that means that there is more than 30 ppm which is very bad. If you and other people in the apartment have headache and nausea that certainly sounds like you have CO poisoning. You can have serious brain damage even with low level chronically poisoning of CO. To be sure you can go to the hospital and take a blood analysis for CO.

If the fire department didn't find anything that not mean that it's safe. If the plumber said you that the problem is fixed that not mean that it's fixed.
If you can, buy another CO detector. And call 911 as much time that it's go off.

About your question, technically what can happen, a lot of think, impossible to said like this, but you have a serious CO problem and should not ignore it.