By all means, fill in any information gaps needed. I would like to know what a 5e ruling to swim across the English Channel looks like. I have absolutely no idea at this point.
That question can’t be answered in the absence of further context. In 5e, you call for a check when a PC’s action has a chance of success, a chance of failure, and a cost or consequence for failure, and you use that context to determine what check is appropriate and what the DC could be. “Swimming across the English Channel” doesn’t have a static DC, nor does any task. To resolve the action, one needs to know the player’s goal, what their character is doing to try and achieve that goal, and what the consequences might be for failing to achieve that goal.
Trying my best to use your example, winning the princess’s hand in marriage seems to be more of a long-term quest objective, rather than the immediate goal of the action. The goal indeed seems to be to reach the opposite side of the channel, and the approach seems to be swimming across it. Now, resolving this action is going to take some DM judgment. Do you think the approach of swimming across has a reasonable chance of succeeding in achieving the goal of reaching the other side of the English Channel? Personally, I’d say it seems... possible... but not if the character is wearing metal armor, and extremely difficult in any event. It definitely has a chance of failing.
What might the consequences be for failure? You could just say they drown if they fail, but that only seems likely to insure the players decide not to even attempt such a task. Since the long-term objective seems to be to marry the princess, and she might marry the brother if they fail, I’ll make the assumption that the player is trying to reach the other side to stop the wedding to the evil brother. In this case, time might be an appropriate stake, and failure could mean progress with a setback - you make it across the channel alive, but not in time to stop the wedding.
So let’s say the wedding to the evil brother is going to happen in 24 hours. At its narrowest point, the English Channel is 21 miles across. In overland travel, characters can cover 3 miles per hour at a normal pace, 4 miles per hour at a fast pace, or 2 miles per hour at a slow pace (and we’ll be dividing those numbers in half since swimming costs twice the normal amount of movement). Let’s assume the player doesn’t care about moving stealthily and isn’t worried about being ambushed on the journey, so they take a fast pace. That would mean they could cover the 31 miles in 15.5 hours. Plenty of time to make it to the wedding, unless the waters are rough, in which case you need to succeed on a Strength (Athletics) check to make progress. But, checking your example, you said average waves, no wind, no rain, so that doesn’t seem necessary. The character can make it across in 15 hours.
Also, just noticing you also said no time limit, so I guess my stop the marriage scenario doesn’t actually work. But it was still an interesting thought experiment, so I’ll leave it in. At any rate, we’ve established that with no time limit and no complicating factors such as rough water, a Strength (Athletics) check is not needed to make progress. But also remember that traveling for more than 8 hours in a day constitutes a forced march.
With a Forced March requiring a Constitution saving throw for each hour of travel beyond 8, the character is going to need to make 7 constitution saving throws. Also, the DC starts at 10 and increases by 5 each time. And on a failure the character gains a level of exhaustion. Depending on the character’s modifier, they’re likely to reach the point where the DC is higher than they can possibly roll after maybe 3 or 4 hours. So this character is taking at least 3 or 4 levels of exhaustion, and needs to make 3 or 4 Con saves at DCs 10, 15, 20, and maybe 25, taking an additional level of exhaustion for each failure (resulting in death if they fail 2 or 3 times).
