Are you able to keep him from wiping his nose on his sleeves when he's at home? Because when my boys were that age no matter how much my husband and I tried or how many times we told them, those long sleeves are just so gosh darn convenient that they didn't care about the 4 or 5 boxes of kleenex I had scattered around the house.
If he was really congested when you picked him up, he probably has a bad cold. Does he already know how to blow his nose? My 4 year old is starting to get pretty good at it, but my 6 1/2 year old is really still learning. If he doesn't know how to blow his nose, no amount of wiping is going to help him be able to breath.
I would be willing to bet that the teachers wiped and cleaned his nose several times today, but if he has a really bad cold (and it sounds like he does), it won't take long before his nose is a mess again.
It is frustrating, but it's more colds at his age that are frustrating. Daycare workers can only do so much. He will learn how to deal with this more, but he's so little.
My 4 year old attends daycare, and they do the best they can. But at home I have trouble keeping up with his nose if he has a cold ... and I only have 2 kids!
I get that you're upset, but I really don't think this is a battle worth fighting. Remind your son to use a kleenex (or ask for one) when it bothers him. It's something he is capable of doing, but if he's anything like my boys it will still take some time.