I studied Computer Science at Warwick so I provide some comments about the two courses. I studied with many Discrete Mathematics students and quite a few Mathematics students. The two courses are quite different, even compared to other the joint courses run by Mathematics.
The Mathematics course is unsurprisingly run by Mathematics. I think it’s accurate to say that it’s one of the strongest courses in the UK or at least that’s the impression I get from the people I know who picked it. There is the chance to study some computer science modules in each year but it is not really common. In my first and second year the mathematics students were prohibited from studying the first year programming module because the person teaching some of it also taught a programming module in Mathematics. If you can study the programming module based in computer science now then it is better and will set you up for the ones that you could choose to study in later years. The Mathematics students who wanted to learn programming but were prohibited in my year picked the computer architecture modules instead. That module is by far the best computer science module in first year and teaches you enough C that you can pick up other languages quite easily. It also helps if you would like to go into working with hardware, engineering or software-hardware co-design in the future. In second and third year you would have to do programming modules in Mathematics (e.g. Mathematics by Computer) or be at the mercy of computer science staff letting you onto their modules. That might not sound ideal if your goal is to learn programming (crossing your fingers for two years doesn’t sound like a good plan) but Mathematics students from Warwick will go into software engineering positions when they graduate. I don’t think you would be at a big disadvantage at all. The Mathematics course is very strong and there is enough programming to put you in a good position, assuming you pick the right modules.
The Discrete Mathematics courses is mostly run out of Computer Science. Your personal tutor would be in Computer Science and you would go to the course directors or link people in computer science if you have problem or need help with the course. One of them likes Discrete Mathematics students so much that he ignores Computer Science students (literally did not even listen or respond to them) so you will probably get help when you need it. You get less support for the highest credit module in first year (Analysis), which I know a lot of Discrete Mathematics students still aren't happy about. You will get all the computer science and programming modules you could want to study but you might not like them as much as you think if you are splitting your time between computer science and mathematics. Some of the people I studied with would completely rule out software modules because they knew that everyone else (computer scientists) would probably be more better at them because they get more practice in that area. I think Discrete Mathematics is a good course but it is important to think about whether you actually like the content. In my opinion you can learn to program on Mathematics or Discrete Mathematics but studying Discrete Mathematics is more specialised than just adding programming. I think a Discrete Mathematics students would usually be less effective at programming than a Computer Science student and usually less effective at mathematics than a Mathematics students so you really have to want to be good at the specific area of Discrete Mathematics if you choose to do it.
It is important to keep in mind that this is based on my experience and that yours could be different. If you have any specific questions then I can try to answer them too.