My advice is below...
Are big law firms still old boys' clubs? Far less than they used to be, but they remain highly elitist – that’s their business model.
Getting a training contract at these places boils down to two things: 1) having top grades and 2) being their type of person (i.e. not a total nerd, presentable, well-organised, some breadth of interests, hard-working, not too much personality).
From my experience, big law firms actually aren’t all that bothered about where you went to uni. It’s just that, for whatever reason, the public school plus Oxbridge/Russell Group route is great at churning out the 'their type of person'-types that they’re looking for.
Perhaps a better way to look at it is: would you fit into a dry, tough, yet high status, working environment? Is that the sort of place you'd enjoy spending your post-Open University days?
If so, go for it.
If not, recognise that the big firms get disproportionate publicity among students because of the resources they devote to marketing their graduate recruitment programmes – and start exploring the many alternatives.