The spell Enlarge/Reduce can be used as analogy.
The content on Wild Shape in the PHB and Xanathar’s Guide to Everything has no information about the possibility of insufficient space.
The spell Enlarge/Reduce does, however. The Enlarge part of the spell states, inter alia:
If there isn’t enough room for the target to double its size, the creature or object attains the maximum possible size in the space available.
Of course the part about objects is irrelevant and the increase in size depends on the monster wildshaped into. But an analogy can be made for larger size wildshapes since this is an size-increasing and magic effect like the spell.
Of course, a change in shape occurs as well, making it closer to Polymorph and True Polymorph but those spells do not have a limited space clause.
Launching away creatures would definitely be a house rule since that kind of effect would need to be stated in the rules for Wildshape. As GM you can of course decide to make Wildshape work that way.