Hi there and
to the forums!
Were both of the budgies obtained at the same place? I'm asking because even if they are adopted at the same time, they could be carrying a sickness from where they were previously housed so it's important to follow quarantine in all cases except when budgies are from the same place within one week of each other.
Angie is absolutely correct that the very first step is to separate them. Since I'm assuming none of them are hand tame, it may be difficult to move them, but if you use a light cloth to gently pick one of them up, s/he won't associate your hand with danger in the future.
It does sound like s/he's sick, and Kristen has provided great information on how to take care of your little one while you wait for an avian vet appointment. It's imperative you get him/her to the vet as soon as you can as this illness must be in the late stages if s/he's so weak s/he is on the floor.
Bullying does seem to be a problem as well, but I also don't believe that it is the main cause of the problem. However, it is definitely a contributing factor, so that's another reason that separation is essential other than quarantine. When quarantine has finished and the little one feels better, introducing them gradually in neutral territory is essential so neither feels territorial. If they get along, after a while, they can be caged together. However, it is not guaranteed that even after this, their personalities will be compatible, so they could need to be caged separately afterwards. Check out this link for more information:
Be sure to also read through the rest of the forum articles and "stickies", as they will keep you posted on the very best of budgie care and practices!
If you have any questions after reading through everything, please do ask as we'd love to help.
Hope to see you around and please keep us posted on your budgies' conditions!