Residents described hearing loud crashing noises and strong winds at about 02:30 GMT.
Wessex Primary School has reported damage including a hole in the roof, dislodged canopies and solar panels upended.
The Met Office said it could not "confirm or deny" reports of a tornado.
Some of the damage from last night's 'tornado' in Cox Green pic.twitter.com/YSV9HCSVtz— James Harrison (@JamesHarrisonBM) January 11, 2016'Rubble rain'
Resident Josie Pass has described seeing plastic sheeting "flying around in circles" outside her bedroom window.
She added: "The noise was deafening and the house was vibrating, it must have lasted for 30 seconds.
"The windows were shaking and my first thought was that we are about to be hit by an aircraft.
"There was a huge bang and the sound of rubble raining on the roof," she said.
Leanne Norcott, who lives in the town, was woken up by a noise and also described seeing plastic sheeting "flying around in circles".
Solar panels have been dislodged and canopies damaged at Wessex Primary School in Cox Green, said head teacher Nick Stevens.
The school has remained open as usual.
The Met Office said there have been observations of heavy showers and locally strong winds in certain areas, but a tornado was "highly unlikely" and may have been squally winds, which can have a similar sort of effect.
The area between Reading and London is the "most likely place in the UK to be hit by a tornado", according to research by University of Manchester in June 2015, which revealed there is 6% chance every year of a tornado occurring in the area.




Comment: Other incidents of 'rare' UK tornadoes in the past year include those in: West Sussex, Northampton, Somerset, Newport, Northern Ireland and Wales.
Just a small selection of other 'rare' tornadoes from around the world in recent months, include those from the United States, Northern Ireland, England, Australia, Hawaii and India.
See also: SOTT Earth Changes Summary - December 2015: Extreme Weather, Planetary Upheaval, Meteor Fireballs