You continue casting probably, but the DM can say otherwise
Let's try not to put the horse before the cart. See the Basic Rules, chapter "How to Play":
- The DM describes the environment
- The players describe what they want to do ...the DM decides what happens, often relying on the roll of a die to determine the results of an action
- The DM narrates the results of the adventurers’ actions
"Spending actions" is the game mechanics, the DM uses them to determine the outcome. You spend an action because you're casting a spell, not vice versa. The same way we use "Attack action" and "attack roll" to simulate combat, for instance — you make these rolls because you're attacking an enemy.
That's why this comment makes sense:
It says you must spend your action each turn, but it does not say that if you do not spend an action that the spell fails
"Casting a spell" takes place in the game world. "Action" does not exist in the game world. We use game mechanics (actions) to model in-game events (spellcasting).
Let's take a specific example. The Wizard is channeling a spell. Suddenly, goblins attack. The Wizard is busy and distracted, so he is "surprised". In the game world that means he is not ready for the fight yet, maybe he is not aware of the goblins at all. So, what does he do instead? He either fumbles, or continue the channeling. This is what happens in the game world. This particular situation is quite rare and is not described in the rules, hence, requires DM's adjudication.
5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While the rule zero has always applied, 5th edition chooses not to explicitly codify many things. For example, your DM may ask you for a concentration check. That's why the game requires a DM, after all.