My 4week old has a blocked nose been a week also has conjunctivitis type stuff on his left eye slowly moving to right?

Adenovirus : Causes respiratory symptoms and comjunctivitis. Which would be common at this time of the year. If we’ll no treatment required but is contagious until eye redness gone which can be weeks. Needs review if breathing troubles or vomiting

Viral infection : Viral infection can present as runny and blocked nose . It can also produce watery eyes and sticky eyes too .

Checkup with GP: A week is a long time for a newborn to be experiencing upper respiratory symptoms. If they still haven't resolved I would recommend a review with your GP as soon as possible.