The post I prepared last year about this medieval shrine can be read at Our Lady of Grace in Cambridge
I have now found in addition the link to the very detailed and interesting account of the medieval Cambridge Dominican priory in which the statue was housed in the Victoria County History of Cambridgeshire. This account of the house can be read at Friaries: Dominicans, Cambridge
The article states that the first reference to Our Lady of Grace is in 1515, which suggests that it was fairly new as an object of devotion at that time, although it clearly became popular and attracted benefactions in the following two decades. The article also gives inter alia more about the later career of Prior Gregory Dod/Dodds.
It contains the suggestion that a late fifteenth century statue now in the amazing and spectacular 1885-90 Catholic church in Cambridge, Our Lady and the English Martyrs, might in fact be the original statue of Our Lady of Grace from the Dominican friary. This is also discussed in the Wikipedia account of the church as follows:
The church houses a statue of the Blessed Virgin Mary which is believed to date from at least the 15th century. Although evidence is inconclusive, the statue, which was discovered at Emmanuel College in 1850, is supposedly the same statue that Cromwell ordered to be removed on 30 August 1538. Experts disagree over the exact date of the craftsmanship, with the Royal Commission on Historical Monuments dating it as "mid-16th century"
Our Lady of Grace in Cambridge Pray for us