This depends on a lot of factors - the type of banana, your location, and gardens in cities are warmer than those in rural areas, and of course how low the temperature falls, and how long a particular sub-zero spell lasts.
Wrapping up only offers limited protection. If weather is severe and prolonged, then wrapping up and/or moving into a cold greenhouse, will not be good enough.
The roots of plants in pots will get colder than roots that are in soil. Bringing a pot into a greenhouse can reduce that effect slightly, but if it's a cold greenhouse and the temperature drops below freezing, then the roots can still freeze.
One of the most popular bananas is Musa basjoo. This is relatively hardy and may survive outdoors, with some wrapping up. But keeping this plant in severe conditions is still difficult.
The attractive banana with wide leaves, Musa ensete, is not as hardy as basjoo. Ensete also has a very attractive red-leaved variety. The red variety is even less hardy than the green variety. It would be difficult to keep a red ensete without heat.
Folding the leaves up, before wrapping the stem, can give additional protection (but it's not easy).
..., should the pot be wrapped in bubble wrap
Bubble wrap can help. But it is important to understand that bubble wrap, or fleece, merely act as insulators. They reduce the speed at which the pot gets cold. So if the cold weather is prolonged, and the temperature doesn't rise above freezing during the day, then they will not prevent the pot from freezing.