The BE expression, "I am reading history at XYZ University, is archaic, localized English vocabulary used by Oxbridge (composite word - universities of Oxford & Cambridge) undergrads, especially by those who have "come up" from the UK's elite grammar schools and the super-elite public schools (fee-paying) such as Eton (alumni Princes William & Harry), Harrow (alumnus Sir Winston Churchill), Stowe (alumni David Niven & Sir Richard Branson) et al.
The specialized usage never really caught on with the postwar development of redbrick universities in the UK. Its use at Oxbridge ebbed and flowed at a time when state educated student numbers from comprehensive schools narrowed the gap with their grammar and public school counterparts and almost drew level in numbers. Reading history or whatever has been, to some extent, replaced by "studying" history. This was almost always the case with redbrick universities. The OP's "I am reading [subject] at X university" is the present continuous, but the simple present tense as in, "I read [subject] at X university", was never written or said but replaced by - as a matter of linguistic convention - "I study [subject] at X university".
If one were to write, "I read history at Oxford, then this would be understood as the past tense. The pronunciation of "read", but not its spelling, would change to "red", as in the color. (Wikipedia)