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.
