I just went through this with my 3 month old. Snotsucker/nasal aspirator to clear out mucus, saline drops to help break up mucus. Check their temp: if they have a fever because theyre so young you do need to have them checked by a doctor to make sure its not something else. If they're having trouble breathing or you're worried about them having trouble breathing: emergency room. I work in an ER, so please, if they're not having trouble breathing or having a fever, don't take them to emergency. There's nothing we can do if it's not a fever or trouble breathing aside from maybe a nasal swab to check for certain viruses that will take a couple days to come back with results.
Nasal spray, saline spray. If baby has a temperature best to take them to ER. My son got covid at 2 months old. And we did have to take him to the ER because he was running a fever, and it wouldn't drop. Just check the temperature. Another solution is run the shower as hot as you can get it. Make it pretty steamy in the room. It will help lose the mucus. Or try a warm bath, if baby doesn't have a fever
My 2 month old has a cold at the moment, luckily he’s not too unwell so I’m just feeding him on demand to keep him fluids up, lots of sleepy cuddles, checking his temperature a couple of times a day and then when he’s all stuffy we are using the saline spray and snot sucker. There’s not much you can give them at 2 months as it’s all 3 months+. You could give calpol if they are really whingey. If baby is struggling with breathing or sucking in under the ribs make sure you go straight to a&e. Wishing your baby a speedy recovery x
Emergency room? If your 2 month old is coughing that could lead to problems breathing. I would go to a hospital