Is it possible for me to fly from France to the UK with just an ID card ? If this is possible which airlines will allow you to fly with just an ID card ?
If you are from a country whose citizens can enter the UK with a national ID card in lieu of a passport, then yes, you can fly to the UK with your national ID card and no passport. Those countries are the countries of the EU and the EEA, plus Switzerland. https://www.gov.uk/uk-border-control/before-you-leave-for-the-uk.
Note that the "ID card" here is a national ID card that must indicate your nationality. A driver's license or other form of ID does not meet the requirement.
If you are flying to the UK with a national ID card, all airlines will allow you to board with that ID card.
If you are a national of another country, you cannot enter the UK without a passport. (Depending on your nationality and the purpose of your trip, you may also require a visa.)