The Duchess is associated with the Adjutant General's Corps. I think that the second badge in Baxter's post is from the Adjutant General's Corps of which The Duchess is deputy Colonel-in-Chief.
Here is an example of the corps' badge
As for the first badge, it looks to me to be some sort of presentation brooch from Queen Elizabeth (later the Queen Mother).
I say this because the "II" is missing from the monogram which would be there if is was something relating to the reigning Queen Elizabeth and I am pretty sure the badge/brooch is not from the reign of Queen Elizabeth the First.
Here are a couple examples of other "ER" presentation badges/brooches which have some similarities.