I think if you tried doing this 30 years ago it would be harder but now it would be much easier. The below might sound mean but it's not meant to be, I really wish it wasn't so obvious but I don't have any control over social policy...
First of all, you could pick off the smokers by the smell or (if long term smokers) the skin damage. It's not a perfect system but the numbers of middle and upper class people smoking have dropped dramatically in the past 30 years.
Second, you could probably look for signs of drug use. Again, not a perfect system but it'll lower the odds of someone being working class if there are signs of drug use, especially cocaine which seems to be a more middle class hobby.
Third, hair will still give things away to some extent. If someone has badly coloured hair (especially the copper kind that looks like an attempt to not be fake) that's a pretty good sign of them being working class.
Fourth, tattoos. Middle class people are starting to get them now but located in certain places like the lower back or upper thigh, oversized or bad quality it will still give people away. Middle class women are more likely to only have a small tattoo on the upper back or something flowery on a foot.
Fifth, nose rings. Massive giveaway. Nose studs aren't, but nose rings definitely are.
Sixth, general health. We live in an age when millions of people have bad diets either because they rely on foodbanks or on cheap food full of hydrogenated fat. I encourage people to look at the differences in the quantities of fat in supermarket budget and supermarket finest ranges to see what I mean (cheap burger cheese slices in ASDA have around three times as much fat content as their standard range for example). I could probably spot the poorest people because they were gaunt and pale and the second poorest group would be slightly overweight (without red "middle class face" from excess consumption of wine) through no fault of their own. People in particularly good health I could probably accurately identify as being at least professional middle class.
Seventh, bad teeth. NHS Dentistry is in a mess at the moment. People with bad teeth would probably be poor to lower middle class.
The system would by no means be perfect but I reckon I could get it right more often than I got it wrong.