Having just graduated with a physics Bsc from an upper Russell group (just under that 1st class but only because I hardly worked in the third year, having burned out - I also have just under 600 UCAS points with AAAA in A Levels Maths, Further Maths, Physics and Chemistry), Id like to offer my 2 cents here.
Your choice of study definitely does matter but so does your sociability. Your choice of study does matter because as I discovered the hard way, even within STEM, employment prospects are, ceteris paribus, hardly equal. I lost count of the many engineering and software jobs I applied to in the third year (with a predicted 1st class) and I couldn't even make it past the filters / HR. Got 3 interviews for an engineering job and two software consulting jobs, and got rejected because I lacked that specific piece of paper with engineering or computer science written on it, but also because I lacked the skills they were looking for.
At the end of the day, employers do not care about whether you can solve the three-dimensional Schrodinger equation for a hydrogenic atom, or whether you can compute the Riemann tensor of a conformally-flat metric. The only thing this demonstrates to them is that you have some capacity for mathematical and abstract thought. The problem for physicists is that there are many engineering and computer science graduates who similarly have such aptitude for maths/abstraction as us, except they also acquired all the requisite knowledge and useful skills involved the fields of engineering and computer science. Why would they take the risk - in fact, waste their money and time - and train you when they could simply hire those people?
All of those platitudes about the supposed 'versatility' of pure science degrees are lies. And I am the tip of the iceberg. Consider this physics PhD who could only find work in a cell centre.
http://www.dailymail.co.uk/news/article-2304096/Graduate-physics-PhD-31-fell-death-block-flats-taking-job-centre-qualified-for.html If you read Physicsforums.com, you will find maths and physics PhDs posting there who have been unemployed or underemployed (e.g. physicists working as bartenders, pizza delivery men etc) for significant periods of time (I will not but can name posters on there).
The continued specialization of the 'applied science' industry (engineering and IT) means that that even within STEM, your choice of study is becoming legitimately subject to discrimination. Just because you have studied physics or maths no longer means you have a decent shot in the engineering and IT industries anymore - yes, you can still enter the industries but only if you acquire the requisite skills to become competitive on your own, and your pure science background does not directly help you in this endeavour whatsoever.
On the other hand, a degree is also a piece of paper in the end. I personally suffer from social anxiety disorder (check out some of my past posts for more detail on the problems this has caused me) which means a significant proportion of the employment sector is simply out of the question for me. Sociability is in fact important, those business, political science etc graduates who found jobs in sales, marketing and other roles do in fact deserve it insofar as they are apt for those jobs (and academic ability is not the main criteria there). No matter how hard I work, someone like me can never become competent in certain roles (requiring a lot of sociability and there are a lot of them).
I've said it before here - I still love physics but it is a bittersweet feeling because I paid exorbitant costs - a huge £30k+ debt and 3 years of my life - only to reap such things as 'intellectual satisfaction' (which I could have frankly attained on my own by studying physics as a hobby anyway).
I understand that the climate of the economy plays a big role in my troubles but my advice to pure science aspirants is now that the ubiquitous platitude they all tell you "study / do a degree in what you love most" is not very good advice.
The reality is that it would be better to make a trade-off between passion and career prospects. If you like maths and physics, go for engineering or computer science. If you like maths but don't like physics, consider quantitative areas of finance such as economics, accountancy and actuarial science. If you like chemistry and biology and if you can, consider medicine, pharmacy or optometry.