First job in industry for a pure, pure mathematics folk

I disappointed a number of interviewers when I had to explain that model theory was not much to do with mathematical modelling...

I suspect you overestimate the amount of maths needed for quantitative work in industry. You have had a training in structural understanding of hard numerical or abstract problems, and this is valuable in itself. Quantitative people are rare.

For me it was easy to learn on the job all the statistics needed for work as a hedge fund analyst: really an introductory textbook on econometrics covered everything general, and for specifics the tools used by the firm were in front of me. In effect you are given a set of levers, and it's quick to discover what they do. Basic reading on the efficient market hypothesis, say, would have done me far more good at interview and for the first months of the job than boning up on Ito calculus.

So you really needn't hamstring yourself by saying you'll need training before you can start.

Further, I'd suggest it's a mistake to be hung up on "using maths". There are real-world problems that you, as a mathematician, are able to get to grips with better than everybody else. Real problems are worth solving in their own right (especially if you find more socially valuable work than the financial industry) and it's this that gives them interest, rather than the particular tools you use to crack them.

I wish I could give you a more optimistic answer, but unfortunately I don't think there isn't much out there for pure math besides academia. Here's what I can think of offhand:

1) Government labs and research facilities. Defense is huge, of course, and cryptography in particular loves mathematics. You may be uncomfortable with the moral nature of the work, the security requirements, relocating, etc., but it is a sector that aggressively recruits and hires theoretical mathematicians. Also, having a PhD is considered highly desirable there, while it often isn't elsewhere in industry. (The situation may differ in the UK versus the US.)

2) Industry labs. Bell Labs in its heyday is probably the best example here. Unfortunately, there are few industry labs left; in the United States, for example, the behemoth tech companies like Google and Microsoft have genuine pure research departments, but that's about it. There are lots of tech-based start-ups that would be amenable to conducting applied research on specific topics, but there's little support for general, abstract research.

3) Start your own start-up. It would allow you to work on something that you genuinely enjoy, but it's probably not a great idea if money is an immediate concern.

4) Finance companies, software companies, etc. like to hire mathematicians. They prefer physicists (probably for all the data analysis involved in it), but they do look upon a mathematical background with favor.

Those are the options for pure mathematicians who want to continue to work as pure mathematicians, or at least invoke their pure mathematical background. You shouldn't feel obligated, however, to do so. Honestly, you should be able handle even the applied math in industry without any difficulty. It's not that such positions are unchallenging; rather, it's that there really isn't much high-level math involved in most industry work, especially in the positions you're likely to be applying for so soon after finishing your degree. Although industry likes people with quantitative and abstract reasoning skills, there's not much market for advanced mathematics specifically. Even abstract computer science, for example, isn't used as often as you'd expect at a software company. As such, companies are often reluctant to pay for employees to take classes in more abstract subjects. They will often help with obtaining degrees, but that's not helpful if you're finishing up a doctorate. I'll also point out that the cultures in most industry places are very different from math PhD programs', and it might be difficult to switch jump from marketing yourself as an attractive research candidate to marketing yourself as an attractive employee effectively.

On the other hand, one advantage you have is that your background in math shows that you're capable of picking up new and difficult concepts quickly, which employers do like. Even if you don't have, say, the particular PDE class a recruiter is looking for, you can demonstrate that you could do the research on your own and pick up the material independently if it comes up over the course of the work. There are companies that are willing to invest in someone who is clever and talented, yet doesn't have the exact transcript or background they're looking for. (It's nontrivial to find such companies, but they do exist.) Another possibility is to find a company that works on something you're interested in (e.g., machine learning), take a more entry-level position even if it's only tangentially related to your actual topic of interest, and work your way up the chain.

Or, more briefly: It's hard to find a job in industry that involves a lot of math, but it's not hard to find a job in industry in general with a pure math background.

As a graduate in pure math after a masters from Cambridge with top grades and having recently come through the road of getting a great first offer from industry I feel obliged to give my advice.

Zero: if you're indeed extremely smart and is motivated by money (who isn't to some degree?), you might like to try for things like Jane Street, Optiver or Citadel which doesn't need too much/any experience. If not...

One: if you don't have much previous experience in industry, you must expect the process to be very painful and be mentally prepared. I had over 10 rejections before my first offer (and that's rejections after getting through the first stage of the assessment process, the number's more like over 20 if you count the submitted applications). At first I felt depressed because I had spent so long learning so many difficult pure maths concepts only for it to all go down the drain in terms of industry requirements. Because I had attended a careers presentation by a PhD graduate who said exactly the same thing, it made the experience much easier to handle. When you get rejected always ask for feedback and ask yourself what you could have improved. I think this mental aspect is most important, don't fool yourself into thinking "I've got a PhD so I can walk into a great job in industry without extra effort". You have to do something extra.

Two: it's good that you mentioned data science. That's what I ended up focussing on as well! One great thing is that you can take free courses online. E.g. the Coursera machine learning course, and that takes only two or three days to complete if you do it full time. Also you can prove your skills on Kaggle which seems to be industrially recognised. Other MOOCs of note: Udemy, Udacity, KhanAcademy. The former two cost some money but way less than a masters say.

And good luck finding your dream job!