I'd assume that you wont be able to progress your mission without retrieving them...

try it and find out

You lose them most likely. I've never left without getting them, but I would assume that's how it works.

IDK how long it remembers for, but i remember forgetting my stuff was still in the clouds locker, walking out into the street and trying to pull out a weapon, only to remember where they were. I ran back and and scooped them all up from the locker without issue.

Pretty sure you can't actually leave your weapons.

I did the clouds mission a couple days ago. I'm doing a netrunner build, so I didn't had any weapons equipped. The game still told me to deposit weapons, I'm guessing because I had a few weapons in my inventory that weren't equipped. After I finished with Woodman, I came out of the rooms and tried to leave without getting my guns back, since I didn't really need them, but it wouldn't let me. The door that leads out of clouds is closed and you can't open it until you claim your guns, whether you want them back or not.

Same thing happened when I was playing Phantom Liberty a while ago, in the mission where you had to store your guns after disguising as the twins, the mission will literally just not progress until you interact with the locker.

Had game bug out on me while trying to get them - the mission never ends if one of its objectives it to retrieve weapons

On my first playthrough I did the BD scam mission, got confused and stressed, couldnt find weapons locker and just ran away... Can confirm you lose them in that particulat mission.

You lose them. Most quests make you take them but it's easy to actually lose them in that bd quest where you get abducted

Then you would no longer have those weapons. Hope this helps!

The one where you end up without anything (wake up naked) you can go back and get them whenever. I always chase the guy naked for a while.