It will be swinging its head left and right to get a proper view of what's going on.
Birds use both eyes to pin point the exact distance to make a landing. and the wider setting of the eyes on the head (either side for prey species) means they have a wider visual surround hmm panoramic view if probably a better description.
Birds don't blink like we do either they use their third eyelid to blink and at speeds the human eye has trouble detecting.
https://www.britishbirdlovers.co.uk/bird-brain/do-birds-blink
Read this it explains it better than I can but you will read in the third paragraph down
"Small birds can blink so fast with each head movement that human eyesight is unable to register it." That's why you don't think it's blinking as it's doing it so ruddy fast. The eye that is now swollen and closed is probably the issue in the location of where to fly to to land. Being young and inexperienced takes time to get a bearing, clumsy as it's newly fledged and half of its distance location switched off as the eye is swollen.
If you ring your vet they will take it in and they can claim the money they spend on treatment back from the RSPCA. They will more than likely also know of somewhere for it to go and a lot of the bird rescue places will go and collect from them.
Wildies are known to be tougher than we think. There are lots out there surviving and thriving with missing legs, missing eyes etc we just don't notice them as much as we don't pay attention to things blipping around all the time, it's just when we focus on one particular one or attention is brought to one that we notice.