There seems to be a surprisingly fair amount of specific information about the 1091 tornado out there (EX: its wind speed, what it damaged)

No, we definitely do not know its wind speed. What we have is Michael Rowe's estimates of tornado strength on the TORRO scale, based on his reading of chronicle accounts in assorted journal articles from the 70s-90s. He puts the 1091 London tornado at "perhaps force T8." This is an F4 or EF5 on the Fujita/Enhanced Fujita scales you might be more familiar with.

According to the scale:

Severely devastating damage. Cars hurled at great distances . Wooden-framed houses and their contents dispersed over long distances, stone or brick houses irreparably damaged, strong damage to skyscrapers.

Not a lot of cars in the Middle Ages, last time I checked. So where is Rowe getting this?

The foundational account of this tornado (turbo--whirlwind) tracks back to the chronicle attributed to Florentius (Florence), a monk at Worcester. While his actual role in the Chronica is debateable--most scholars assign the actual text to another monk named John--the relevant data here is that the account was placed in the chronicle long after the fact and not in London. (Although if the actual monastic writers were working from earlier notes/a lost text is, well, lost. William of Malmesbury gives a very, very similar account to 'Florentius', which might support the existence of a shared earlier account)

According to the account (Latin here), a strong "turbo" marauded through London, damaging or destroying six hundred houses and many churches. Rowe takes the 600 number literally--which he should not--and assumes "destroyed", leading to his high estimation of tornadic strength.

Additionally, two people died in St. Mary's church, presumably from the roof collapse/destruction. The Chronica attests that several of its beams were sucked up into the whirlwind and flung back to the ground so hard they were reportedly driven several feet into the earth. ("Several"--the account actually claims, oh, 20+ feet, which Rowe also seems to take literally.)

I'm skeptical of Rowe's estimates of the strength of various premodern tornadoes. However, it's fun to read about the different storms, and intriguing to see what he--mostly based on the work of Charles Ernest Britton--interprets as "tornado" based on medieval accounts. I admit, sometimes it seems a little bit like modern-day fundamentalist Christians looking at Bible apocalypse accounts and saying, "See? That's DEFINITELY a helicopter!)

I'd start with Rowe, "Work of the Devil: Tornadoes in the British Isles to 1660," Journal of Meteorology 24 (1999): 326-38.

The two main sources are William of Malmesbury and John of Worcester, neither of whom were eyewitnesses, but they're the earliest information we have.

According to William:

"A clash of conflicting winds, coming from the south‐east, smashed over 600 houses in London on 17 October. Churches and houses, enclosures and walls were left in heaps. The violence of the winds produced a yet worse disaster: it raised the roof of the church of St Mary-le-Bow, and killed two people there. The timbers and beams were carried through the air - an amazing sight for those watching from afar, but a terrifying one for those standing nearby, lest they should be killed. Four timbers, 26 feet long, were driven into the ground with such force that scarcely four feet protruded. It was remarkable to see how they penetrated the hard surface of the public street, in the same arrangement as they had been laced by the craftsman's skill, being an obstacle to passers-by, they were cut off at ground level, since they could not be removed in any other way."

This was translated by M. W. Rowe in Extreme Weather: Forty Years of the Tornado and Storm Research Organisation, ed. Robert K. Doe (2016).

There is a recent translation of William of Malmesbury's Gesta Regum Anglorum (trans. R.A.B. Mynors, Rodney M. Thomson and Michael Winterbottom, Clarendon Press, 1998), but I don't have access to it at the moment.

The other account by John of Worcester says:

"Moreover, on Friday, 16 October, a very powerful whirlwind from the south-west shook and destroyed more than 600 houses and many churches in London. Raging through the church of St Mary-Le-Bow, it killed two men, raised up and carried through the air for some time its roof and timbers until six of the beams, in the same order in which they had previously held the roof in position, were sunk so deeply into the earth that there was visible the seventh part of some, and the eighth part of others. They were twenty-seven or twenty-eight feet long."

Patrick McGurk, trans., The Chronicle of John of Worcester, III: Annals from 1067 to 1140 (Clarendon Press, 1998)

They differ on the day but otherwise they're very similar - maybe one was copying from the other, or they were both using the same earlier source. There are some other later sources as well, but they were just copying from William, and have no additional details.

Neither William or John actually call it a "tornado" since that concept didn't exist yet, but based on the descriptions of the wind and the damage, modern researchers have concluded that that's what it must have been.

There were some other medieval tornadoes in England - for those, see:

M. W. Rowe, "Tornadoes in medieval Britain", in Journal of Meteorology 1 (1976), 219–222.

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EDIT: Ah once again sunagainstgold is quicker on the draw!

Does anyone know of any credible account from 1091 of the then London Bridge or any bridge across the Thames being affected by the tornado? lots of great feedback here, for my needs --and thanks to WelfOnTheShelf for those direct quotations.