I think a lot depends on what is meant by the various class definitions. What constitutes working class, or managerial class these days? Is someone who supervises a team in local government, or in a call centre 'managerial', for example? Are their direct reports 'working class'?
To claim that, say, a skilled shipyard worker's son or daughter has become a call centre team leader, so has moved classes is disingenuous, in my opinion. It took years to train to be a 'skilled worker', whereas these days, when the majority of people work in offices (once a middle class marker) there is far less of a gap between the achievements of those 'in charge', who are usually not managing at all, but carrying out the executive decisions of others.
People cite heavy industry as examples of strict class divides and lack of mobility, but libraries, CIU clubs (which are now basically drinking establishments, but used to be pushers of temperance and education), the WEA and so on have all grown out of unionisation of workers in heavy industry. The Pitman Painters is another example of how people in very heavy manual jobs were capable of so much more if only they had been given the chance.
I agree that for the Paulas of this world, life is so much more difficult, and think that they should be given credit, not looked down upon, as is so often the case. So often on here, I see posts from people 'subtly' bragging about their grammar school education of 50 years ago, or the 'respectable' university they, or their descendants attend(ed), and this is more evidence of the things that hold back Paula and her friends. It is often not intelligence that stops them from getting into popular university courses, but interviews that ask about things beyond their experience, and application forms that take account of DoE awards, or Operation Raleigh gap years that are way beyond the reach of families in which everyone is working as many zero hours contracts as possible to put food on the table.
We can have as many anecdotes about individual families on here as we like - they prove nothing, really. What they do show, however, is the way in which we all tend to see things in terms of how our lives have worked out. I did X, so there is no reason why others can't do the same.
Also, coming from a working class background years ago probably meant dad in a stable job, mum around to do the chores, a cheaply rented council house and stability. All of these things are pipe dreams for many 'working class' families nowadays.