No, they don't employ proof readers and haven't done so for some 20 years. Proof readers were part of the hot-metal infrastructure.
I've worked in national newspapers for nearly 20 years and there are certainly fewer subs than there used to be and I'd also venture to say that the young reporters taken on often can't write as well as you might expect. This is due to several factors, I think. The first is that training in journalism is much less formalised than it used to be. It used to be very rare for a young person to go straight into a reporting job on a national; now, it's common. That means they've missed out on learning the job on local papers where ancient grizzled subs kick style and reporting procedure into them.
The second is that daily newspapers are hard work - there simply isn't the time to spend with a young inexperienced reporter patiently going through a poorly written piece saying things like "you must check the spelling of names, you must learn the difference between its and it's, this is an ugly sentence, this is a lovely paragraph".
I've flung pages together in a couple of hours flat plenty of times and I know there were things that were not up to standard on it: poor headlines, an ugly widow, a spelling error that I didn't catch. Generally as the night goes on and things slow down, mistakes will be caught and corrected so the nearer you live to London, the later the edition you get and the more tidying-up will have been done.
That's a long way of saying that most newspapers don't employ enough journalists.