No effect other than being close.
The space rules are mum about effects of starting a turn in the same square as another creature.
The creature in the same square is uncomfortably close for combat.
The moving around other creatures rules do point out that you can't willingly stay in that situation:
Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
Since your turn includes a move and you can't end your move still in the same space as another creature. So using your move to travel 0 feet (staying in place) is not an option. Something has to give. Either you have to move the other creature with your abilities or move out of the square.
It could be construed that occupying the same square as another in combat involves restricting ones self in a similar fashion to squeezing through a restricted space. If this is reasonable for your table, both you and the other creature would have disadvantage on attack and dexterity rolls until the situation resolves.