Hi, I'm Dr. Jill. I'm glad to hear mare and foal have been doing well up until this point, but I'm sorry for the unfortunate turn of events. In adult horses, I don't stress right away about a snotty nose, but any sign of potential infection in a foal can turn very serious very quickly. I would not recommend taking the wait and see approach with this situation. It's possible the foal may recover uneventfully, but I don't tend to take chances with them, especially when they're under 30 days of age (the highest mortality period for foals).
I would strongly recommend having your vet out to take a good listen to the foal's lungs and discuss starting a course of antibiotics. I commonly use high dose ceftiofur in foals, or a combination of amikacin and ampicillin, but that's what has been show to be effective in my area for common bacteria. This may be different where you are.
If you would prefer to wait and see how the foal does, then I'd recommend taking the foal's rectal temperature 2-4 times a day (should stay under 102F), closely monitor the foal multiple times a day for nursing, and palpate all of the foal's joints for heat and swelling as well as the umbilicus (common sites bacteria can end up in) at least daily.
Again, I wouldn’t be overly concerned with an adult horse that was otherwise well, but with the foal being so young and with the fact that foals can go downhill very quickly and have poorly developed immune systems, I would strongly suggest calling your vet to come take a look with any potential sign of infection (like a creamy nasal discharge).
Please let me know if this addresses your question or if you have any additional concerns. I would be happy to continue the conversation further if there is anything else I can do to help.
thank u for your reply, I will take your advice onboard