Depends on the fare conditions. If you are on a restricted ticket and miss a deadline, you could be somewhere from SOL, e.g. having to buy a new ticket, to simply being acommodated on a later available flight. Almost always the latter. If you are on a changeable flight, simply change to a later flight.

Your luggage will be checked with you, so not a question of your luggage following later.

Stuff like this likely covered by your travel insurance.