The Post in "Old street pics" mentioned above leaves a few unanswered questions and as more information becomes freely available maybe some light can be shed.
The Birmingham Gazette prints the proposed Inclosure of Lands within the Royal Town, Manor, and Lordship of Sutton Coldfield, in the county of Warwick in January 1827.
On Coldfield Common...the "Setting up of carriage roads and highways"....one of the roads...
Number 4...A Road of 40 ft commencing at a certain bush called Beggars' Bush, near the said turnpike Road, and extending in an eastward direction over the said common, and terminating at the Horse and Jockey Public House. (Jockey Road?)
No 2...a road of 40ft commencing at the east end of the road in the hamlet of Erdington, called Gravelly-lane Road, and extending in an eastwardly direction over part of the said common into the said turnpike road.
The thumbnail OS Map, for around 1882, shows the boundary between Staffs and Warks more or less along the Chester Road and the Beggars' Bush tree is on the boundary. The hamlet of Erdington is quite a distance away and I can't find any evidence that it was once in Staffs, and is hardly ever mentioned in connection with the Bush.
The map also gives the name of the pub as the Bush, but in all references I can find from 1858 through to 1939 it is referred to as The Beggars Bush with or without the apostrophe in both places! The Inclosure shows the apostrophe after the "s."
The advert for the sale of the Beggars' Bush in 1884 states the inclusion of stabling and a Coach-house.