You said you are about to start studying A-Level Maths, so I think it might be slightly early to think about undergraduate calculus books, such as Spivak. I would recommend 'Pure Mathematics 1' and 'Pure Mathematics 2' by Bostock and Chandler. Despite being written a long time ago, these are the best A-Level textbooks I have ever seen; newer books tend to omit a lot of important proofs.
You are also right to point out that high school calculus in the US (such as AP Calc. BC) tends to cover more calculus than is taught in A-Level Maths and Further Maths. Elementary calculus/real analysis topics not covered in A-Level Maths and Further Maths include continuity, techniques for evaluating limits, and convergence and divergence of sequences and series. In the UK, such topics are typically taught in the first year of an undergraduate maths degree.