I cannot find the Welsh dragon as a cap badge but it is used by the Royal Welsh Fusiliers Museum as their symbol. See http://www.rwfmuseum.org.uk/index.html
Below is another photo of HM from the same day as the one posted initially by PJ. It also says that HM was visiting the Welsh Fusiliers, as do a number of other photos from the same day,so I feel confident that she did visit the soldiers on that day.