The intital condition has the elevator experiencing a force while restrained by the cable due to the Earth’s 1G acceleration, and the balloon experiencing a force while restrained by the string due due to the Earth’s 1G acceleration and due to its bouyancy in the 1G field due to the density difference between its internal volume (low density) and the air around it (higher density). In addition there is an air pressure gradient in the elevator due to the Earth’s 1G acceleration; the pressure is higher near the elevator floor.

When the cable snaps and the string is simultaneously released, three things happen:
1) The elevator is suddently in free fall in the Earth’s 1G field, no longer experiencing a force.
2) The balloon no longer experiences either a downward force due to the Earth’s 1G field, nor a buoyancy due to the density difference between its internal volume and the air, because the buoyancy was due to the presence of the 1G field now removed.
3) But the air pressure in the elevator quickly equalizes.

The net effect is that the air moves upward slightly with respect to the elevator walls, carrying the balloon upward slightly. The air then stops moving with respect to the elevator walls, and the balloon stops moving with respect to the elevator walls.

The elevator, balloon, and air (we ignore the relative motion of the person, which is a complex question) then proceed to fall without motion relative to one another until the elevator hits the basement floor. Assuming a rigid elevator structure which is not changed by the collision, at that time the person pancakes on the elevator floor, the air stops and re-establishes a pressure gradient, buoyancy of the balloon relative to the air returns, and the balloon floats upward relative to the elevator walls, stopping only when it reaches the elevator ceiling.