You will almost certainly need Mathematics. EE is more mathematical than many engineering disciplines, so Further Mathematics would also be very useful (moreso than many others in a way, since the complex numbers and matrices content is much more prevalent in EE than others, from an earlier stage). Any EE degree will also have further modules in mathematics - probably several. If a degree doesn't require A-level Mathematics and only has a single token module in Maths...it may not provide the best preparation for working in industry, and would certainly be much weaker for continuing in a research capacity. You may also want to check the degree is accredited by the relevant engineering body - I think EE courses are usually done by the IET.
The long and short of it is, if you think you'd be unable to cope with A-level Maths, you would likely struggle on an EE (and in fact, probably any engineering) degree. While there may be some more "vocationally" oriented degrees in e.g. Engineering Technology, this is not the same as an engineering degree (as if it's accredited there is a core syllabus that the degree much cover, and this includes a considerable amount of mathematics and physics that underlies the core engineering principles). That said, the sorts of topics in A-level Mathematics tend to be more interesting and engaging than in GCSE, so you may well find that the change of pace and style is to your benefit.
Beyond Maths and Physics, anything is fine - Further Maths is preferable as above, or another science, as fundamentally engineering is normally taught as an experimental discipline and having exposure to scientific lab work and writing is useful. Further, while it may seem a more "vocational" area, as above it is a fully academic course - some more than others, but to some extent they are all based on a core academic backbone (as with the vast majority of university degree subjects). Thus, it's better prepration to take traditionally academic subjects over vocational ones, if you intend to pursue a degree (in any subject, but this certainly applies to engineering) - thus, Economics would be more advisable than Business Studies, and Computer Science would be more advisable than ICT. This isn't as critical as the core requirement of Maths and Physics, and isn't likely to affect your application outside of the "top" universities (like Oxbridge and Imperial), but engineering is a challenging degree so it's to your benefit to choose a challenging set of A-levels to prepare for this.
I was formerly studying EE at Exeter, in case you wanted some understanding of my comments