Actually I went through that about 6 years ago at night. I dropped to the floor until it passed scared half to death. I'd probably do the same again. LOL But if I have time and don't panic I would take my camera, hat and robe on my way to the wine cellar.
Never been in one only saw one on The Wizard of Oz, but judging by that if I had time I would take my Important paper suitcase, my favorite leather jacket and my car keys. These are the theings that I took when my house caught on fire(along with my children ofcourse.)
I'd be astonished if it did, London not being home to many tornados.
If it did, however, I'd take (1) my file with our passports, driving licences and so forth in it, (2) my handbag, as having some cash and credit cards would be rather useful if our flat vanished in a tornado, and (3) a large bottle of water.
My back-up hard disk, for both personal family stuff, incl. photos, and work documents, is at a different address anyway, so I wouldn't need to bother with that.
I've been in tornados, hurricanes and fires. In a torando, I have to agree with "ralwus" - i'd stick. Hurricanes and fires have longer lead times and I've fled both. I loaded up my computer, my cat, the significant other, medical supplies, clothes, food and pillows. Then anything else I could fit in the car that meant something. I hope I never have to repeat these experiences, they were miserable.