Flame detectors
Depending on the location of the candle (remote or reachable), and how long it will be unattended for...
A thermocouple in or just above the flame could be cheap and sufficient (depends on if the candle will burn away from the thermocouple and so give a false alarm). A thermistor could be used but will soon fail if overheated.
An IR detector (possible IR phototransistor or even a photoresistive cell) as used in opto cable links could give a useable signal (basically working as an infra-red beam alarm, the candle being the source).
A photoresistive cell could also be used in the visible light spectrum if there isn't too much background light.
If you can find them there are also Ultra-violet detectors (look like large neon indicator bulbs - glass envelope) available, requires something like 60v across them, any UV causing the gas filling to break down and so collapse the voltage - a visible light is also emitted but the voltage is not high enough to sustain the glow without the UV source (too messy to set up at home?). - Yes these are used in industry !
How about a bi-metallic strip (pulled out of an old car flasher relay) heated by the flame, the bending caused could make/break a contact.
Exotic (OTT?) a sample pump pulls a gas sample from above the flame and feeds it to a Carbon Monoxide (CO) detector. The CO will fall when the candle goes out.
HT across a spark gap placed in the flame - the flame will ionise the air and cause a current to flow across the spark gap; no flame will result in no current (a small spark gap will require several kV at a very low current).
A sealed vessel above the flame will be pressurised by the fluid (gas or liquid) trying to expand inside it - the basis of a thermometer. Temperature switches and indicators are available that use this system and it is possible to make one from a length of microbore copper tube as sold for central heating (smallest bore available is best). This is mechanical rather than electronic but something like this will be at the heart of your hot water immersion heater, central heating boiler (and its room thermostat), kettle, fridge, blah blah blah.