Uber would have to pay VAT on the part they take from the driver though. Uber take what 30%? So if you had a £100 ride, the driver takes up £70. The driver will be under the VAT threshold and so doesn't have to charge VAT. However, Uber's £30 they took from the driver should have VAT. I've heard from drivers that Uber adds lots of fees so I wouldn't be surprised if they actually took £36 from the driver and charged him £6 VAT.
That VAT would go the UK government, but it would be offset by any expenses Uber incurs with VAT.
The UK want Uber to add the VAT before it goes to the driver. So the £100 ride would cost £120. Of which £20 would go the government, and the driver not pay anything (as he is an employee).
So not only are they trying to avoid paying less VAT but they are trying to avoid passing on the VAT to customer, making their prices seem cheaper when compared to people that do charge VAT.
But a black cab wouldn't normally charge VAT either, since they are self employeed and would be under the threshold.