Historic England cites ‘The Muslim Institute’ on Brougham Terrace, Liverpool as ‘the earliest recorded mosque in Britain’, whilst Shah Jahan Mosque in Woking is ‘Britain’s first purpose-built mosque’.

Shah Jahan Mosque, Woking

Source: RHaworth, CC BY-SA 3.0, via Wikimedia Commons

In 1887, the journal The British Architect mentioned that W. I. Chambers, Architect “is contemplated building a mosque at the Oriental Institute and University, Woking, and Mr. W. I. Chambers, the architect, has received instructions to proceed with the work immediately. The style selected is Saracenic, with the usual large dome and clustering minarets. The site adjoins the Indian Memorial House to Sala Jung, now in course of construction from the designs of the same architect.”

Sources:

‘Where Was Britain’s First Recorded Mosque?’, https://historicengland.org.uk/listing/what-is-designation/heritage-highlights/where-was-britains-first-recorded-mosque

https://shahjahanmosque.org.uk

‘Shah Jehan Mosque at Woking’, The British Architect, vol. xxviii (1887), pp. vii