Interesting post. Yes, Sarehole Mill was operating in the 16thC and possibly even earlier. Sarehole and Wake Green were both owned by Maxstoke Priory [nr Coleshill], along with Swanshurst and much more in Yardley in the medieval period. They were later owned by John Taylor, of banking fame. before the "Wake Green Estate" part was purchased by Joseph Dyott [hence Dyott Road].
It is interesting that Moseley Wake Green, as shown on the enclosue map of 1843 is mainly to the south of what is now Yardley Wood Road confirming the use of Moseley Wake Green in what was obviously Yardley at the time. As the map only shows common land not yet enclosed it is probable that Moseley Wake Green was originally larger but that piecemeal enclosure over the years had nibbled away at it. Not too much of a stretch to suggest that the land on which Gracewell Cottages/Tolkien's home was originally part of Wake Green [Common]?
One thing is certain, that what is meant by Wake Green has varied over the last century or so. The area around St Agnes Church was called the Wake Green Estate when owned by Joseph Dyott [as confirmed by my house deeds.] and I have seen other locations for it And then there is Moseley {!] Wake Green Common as shown on the enclosure map of 1847. And now we have a BCC map showing it somewhere completely differently - actually this has no historical relevance - not sure what it was purports to show - perhaps you can clarify?
All very interesting. Keep up the good work.