This it is. As far as I know, the rules do not specify how the creature will occupy space after being transformed. In particular, there are few specifications on how to play using thumbnails / grid in the main books.
For the award, as Rykara mentions in the related response
the grid space of a map does not represent the actual space that a creature occupies, it represents the space that that creature controls
So, there are some things to consider: First, enlarging a creature may not require it to actually occupy the suggested space for a larger creature, which is also backed by the DMG (p. 251):
For example, you can use a miniature that has a large base to represent a huge giant. That giant occupies less space on the battlefield than its size suggests, but it is still huge for purposes of rules such as confrontation.
Second, if you are using the optional "confrontation" rules (DMG, p. 252), it would make sense that the space that the creature controls is defined by the way it was faced.
Finally, for your scenarios where the creature is, for example, surrounded, it would make sense for the creatures to be pushed back, if the creature that grows is in fact becoming enlarged enough to occupy other spaces. Also, in your budget, keep in mind that
If a creature is a friend or an enemy, You cannot voluntarily end your movement in your space.
This means creatures can they occupy the same space, they simply cannot do it voluntarily (in particular, they cannot voluntarily enter the space of another creature moving and remain there). Therefore, another possible award in this case (surrounded creature) is simply to allow them to occupy the same space and, in the turns of the monsters, they are forced to move out of space (since they cannot voluntarily remain there). (The character does not have a choice unless the monsters are Small or smaller, since he cannot move through the monsters.)
For my experience1Ultimately, none of this should have a relevant impact on the game, so you can judge how you want and it should be fine if this situation appears in a real game.
1 Just as a reference, I have played many wizards with Polymorphs and Enlarge / Reduce on online tables (with grid / miniatures) and I don't remember once when the occupied spaces would have changed the game. The DM / I only increased the token size as we believed and moved on.