[St Charles Owl]

As I see it, and my facts may not be 110%, but following a fire in a similar block of flats a few years ago, there was a bill in Parliament to make it law for these flats to be fit for living. This would've made it compulsory for sprinklers to be put in.
There were a load of Tories, many of which were money grabbing landlords themselves, David Cameron included, voted to bring this bill down.

Another thing, with the design of these tower blocks, why only one staircase? Shouldn't buildings like this have one of those exterior fire escapes?

There is currently no law on the books that requires existing building to have sprinkler systems in the UK. New buildings over a certain height have to install them but no retrofit law. This has been a known issue for many years and successive governments have passed various laws to improve fire proofing with materials, doors, walls etc but they have always steered away from making sprinkler systems mandatory in existing buildings.

As regards the staircase, thats just the way these buildings were build in the 60s. External fire escapes are not viable in most buildings and in this case would have actually been unusable due to the fire moving up the outside of the building. One fire escape should work as long as it has the necessary one hour long fire protection measure built into it and as long as the rest of the building is up to code.