The email was gentle and kind but made it clear that the final deadline remains the same, something the School-wide and uni-wide emails have also stressed.

OTOH the Equality Act makes it clear that reasonable adjustments must be made on an individual basis, based on individual circumstances. And if your MH is getting in the way of being able to work on you essays, particularly in the unpredictable circumstances of the current situation, then that's something they need to consider and respond to.