My two cents is that the admission to read CS or Math@Oxbridge is becoming more and more competitive for US students as it is increasingly attracting more highly academically qualified candidates from US. Oxbridge may not be willing to increase the offers in the same ratio as many of these candidates will have quality options back home and may choose US schools over Oxbridge. Hence applicants need to have every part of their applications (including PS and reference) in top order.
Interestingly Oxford was the only rejection my S had (in early round), His results are
Acceptance:
UChicago
UMich
UIUC (CS+Math)
Case Western (25000 merit)
Defer:
Caltech
MIT
Rejection:
Oxford