The newly erected building for the University Galleries and the Taylor Institution was designed by Charles Robert Cockerell in 1839 and built between 1841 and 1845. One photographic print by Talbot, held at the Royal Photographic Society Collection (RPS 25154 from the album, 'Talbotypes from 1843') suggests that Talbot photographed the building when it had not yet been completed.
In 1839 the University of Oxford arranged a competition for the design of a new building to house the University's collection of ancient sculpture and other art and for an institute for the teaching of modern languages (The Taylor Institution, also known as the Taylorian). The winning design, inspired by the Temple of Apollo at Bassae where Cockerell had taken part in excavations, was completed in 1845.
The University Galleries now form part of the Ashmolean Museum which moved into an extension built behind the Galleries in 1894. The adjoining Taylorian is a separate institution which houses the Taylor Institution Library and still serves as the centre for the teaching of Modern European Languages at Oxford.
References:
A. MacGregor, The Ashmolean Museum, a brief history of the Museum and its collections , Ashmolean Museum, Oxford, in association with Jonathan Horne Publications London, 2001.