Our baby caught one from the toddler at less than one week old (when I came home from hospital toddler had a cold so must have caught before the birth). We used saline nasal spray to clear his nose and luckily he was pretty much just congested and didn't have a temp or other symptoms. It didn't last long but as I was quite sleep deprived I can't remember exactly how long!
My 7 week old had a cold for about a week, it started as being bunged up and I used nasal spray (calpol brand can use from birth) and then by about day 4 he started to develop high temp so went to gp who were happy that he was all well but advised I could give calpol but just half the dosage just to get temp down
Our little girl tends to be quite congested in the nose (also second born so probably toddler lurgy!) I have found a humidifier in the room seems to have made a difference plus saline spray 🤷♀️
I'm in the same boat with my 9 week old. I've bought a Calpol plug in (non medicated) for tonight as that helped me when I was pregnant and had the flu. Hopefully it'll work and not much longer for both of us