There could be some confusion here regarding the name of the meeting. As far as I can establish, what you are calling the Westbury Street Monthly Meeting met at the Wheeler Street Meeting House in Spitalfields (Shoreditch)London, which was located on the corner of Wheeler Street and Westbury Street, which later became known as Quaker Street (a name which it carries today).
As the Meeting House was not established until some years after the Monthly Meeting was started, the original name may well have been Wesbury Street, only changing when the Meeting House was established. Therefore, any information regarding the Westbury Street Monthly Meeting may well be recorded in the registers of the Wheeler Street Monthly Meeting.
The original Meeting started in 1656 in the upstairs of a house, later moving into a Large tent in the Yard. It was nearly closed several times during the early years by influential people opposed to the Quakers, but eventually a permanent Meeting House was established. The building was badly damaged 1703 and despite repairs the number of members dwindled, and the Meeting closed in 1740. Five years later, the building fell down.
Very few UK Quaker records have been put onto the internet, but most of the original registers have survived. Many have been placed into the care of local Records Offices, and are accessible to the public.However as the Westbury (Wheeler?) Street Meeting House closed in 1740,they may not have survived, if they have survived they may be stored at the Quaker Meeting House In Euston, I will try and find whether they are there.