that's actually a really difficult question to answer - the number you get back will depend on the botanist/taxonomist you ask! Not only is the taxonomy contentious but also the native status is often in dispute too. Clive Stace in the latest edition of his Flora of the British Isles lists 4 species as Native or probably so:
Ulmus glabra - the Wych Elm. This doesn't sucker and has much larger leaves (>10cm) than the others but which have very short petioles (<3mm)
U. procera - the English Elm (paradoxically this might not be native!) - it has more circular leaves which are rough above and often attacked by a pimply gall
U. minor - Small leaved Elm of which 3 subspecies are recognised based primarily on the shape of the mature tree - the typical subspecies has a broad crown while Cornish (subsp. angustifolia) and Jersey elms (subsp. sarniensis) have much narrower outlines. All have small, proportionately narrower leaves which are smooth above.
Similar to U. minor and thought by some to be another variety of it is Elm (U. plotii) - this seems to be endemic to C. England - it has leaves with a rougher upper surface like procera but of the same shape as minor. It too is narrow in outline with the branches drooping.
Hybrids between these various taxa occur although generally sexual reproduction doesn't occur and most have been propagated (and planted in our landscape). This is one reason why there is contention over how to recognise these things taxonomically - they may generally be fairly distinctive but are we just looking at clonally propagated distinct individuals?
Leaf shape and tree shape are of most use to discriminate them - the problem comes that these days we have few mature trees (so we can't tell what shape they'd be) and the leaves on the juvenile suckering shoots, which are what we now largely have in our hedgerows, are not as distinctive as those on the short shoots of mature specimens!
there was a helpful illustrated article in British Wildlife 13:390-395 by Max Coleman back in 2002. Some of these pictures were then used in a quick easy to use guide that the NHM developed for an initiative with the Ramblers, Ancient Tree Forum and others called Elm-Map. Scans of this do still lurk on the internet and I've attached a copy. It just recognised three main groups - Wych Elm, English Elm and the Smooth-leaved Elms. There are obviously some interesting older books on the Elms - particularly R.H. Richens (1983) Elm by C.U.P and the earlier Epitaph for the Elm by G. Wilkinson (1978) which can sometimes be found inexpensively second-hand.
There are also many brilliant images out there on the internet but you always need to be a little sceptical about identifications.
In response to your question concerning native British elms. Yes I would agree it depends on the Botanist more than anything, but generally all would agree that Ulmus glabra is a native for certain; though Ulmus minor and its associated natural variations are possibly of Roman origin. R.H. Richens (elm expert) did point out that the Cornish elm (Ulmus minor var cornubiensis) was found in areas where the British (Welsh speaking) Celts had settled around the 4th-6th centuries. This included Cornwall, west Devon and Britanny. Wheatley elm (Ulmus minor 'Sarniensis') had originated from the Channel islands and may have been a natural variation at one point... though not really proved. The English elm (Ulmus procera or Ulmus minor var vulgaris) is said to be found also in Spain and Italy and is strongly associated in the Romanized areas of the British Isles. It had been long associated with Roman culture here and is much populated near Roman settlements and roads.
I now of no scientists that had done carbon dating on elms to establish if certain species are native or not. But someone must have somewhere. Of surviving elms. Populations of Ulmus glabra are to be found around north and north-west Scotland; south-west Wales and in woodlands across the UK possibly as single trees or small groups. I recorded one huge tree in Stevenage, Herts a few years ago and had seen one or two good trees in Greater London. Ulmus minor survives in Wiltshire and East Anglia; with a good many mature trees near Upminster, Greater London. There are natural variants of the type around Eastbourne, Rye and eastern Kent (namely near the Isle of Thanet) which may have the same origins as those found in area bound by Christchurch in Dorset and Lymne Regis in Hampshire. This was called the Goodyer's elm (Ulmus minor type) and is believed to be by some botanists as the Cornish elm. However there are some distinct features that confirm differences between the two (leaf serration, tree shape etc.). There may be a fossilized version of a Ulmus minor variant that is an ancestor to all the types found from Cornwall to Kent, under the English Channel. Fossiled remains of Ulmus laevis had been found in recent years.
If you require more information please feel free to contact me by email. firstname.lastname@example.org. Pictures: Upnrwilimington1 shows Ulmus procera near Alfriston, East Sussex (most of which is still there). BRdCe-003 shows Cornish elm in Bear Road Borough Cemetery, Brighton and JerseyElmEG1 shows Jersey or Wheatley elm at the end of the street where I live, near Elm Grove, Brighton.