I would echo sentiments above to just select any reputable maths course, as you will inevitably cover at least some "applied" maths and realistically the majority of maths courses will have a majority of applied options anyway (e.g. Cambridge, Oxford, Imperial, Warwick, the latter perhaps less obviously; these of course at the very "top" end of the spectrum but are in general the ideal model for what you want to be looking for...). In general to be an "applied mathematician" you need a good background in what is nominally pure mathematics i.e. at minimum "core" analysis and (abstract) linear algebra at the undergraduate level.
It is better to have a broad and thorough background in all those relevant areas of mathematics to prepare you for a career as an applied mathematician, or potentially a pure mathematician or something else entirely, than to try and pigeonhole yourself into a round hole and find you're a square peg halfway through the course. If you have a specific area of interest in mind (for example mathematical/theoretical physics; theoretical computer science; etc) then a joint honours course in the relevant "other" subject usually gives a suitable background, balancing the more abstract, but still ostensibly "applied" maths options with the core teechnical content of the "other" subject. Some examples are Warwick and UCL's Discrete Mathematics or Mathematical Computation courses, respectively; Birminghams Applied Mathematics and Theoretical Physics course; or Bristol's Engineering Mathematics course, among others.
Given you aren'y sure about what area you may want to end up working on in particular I would again reiterate the comments above of just looking for a good "general" maths course. I would also highlight that the definition of "pure" vs "applied" for school mathematics has no correlation to how these terms are used in degree elvel mathematics; most of what degree level mathematics would refer to as "applied" is similar to what is known as "pure" mathematics in A-level - it primarily contends with general, analytical solutions and being able to rigorously prove that solution is in fact the (or a) solution - or that there is no solution. What is actually referred to as pure mathematics in a degree is much more abstract than anything from the A-level course...but applied mathematics in university mathematics does not mean "we solve a problem and get a neat numerical answer"; that is the realm of engineering and (to a lesser extent) physics.
If you just want to solve equations for problems, physical or otherwise, all day, get a degree in engineering, or perhaps mathematical/theoretical physics; you can apply the many methods in your "toolbox" by the end of that degree to any number of varied academic fields including e.g. biology, economics, etc, etc (many people go from backgrounds in engineering and/or physics to work in those areas). If you want to do mathematics, rather than just use it and thus actually be a mathematician (in the general sense, rather than academic) then mathematics is probably the area you want to look at in all it's varieties (i.e. including pure mathematics, which is a core part of any mathematicians training as stated above).