What if anything are these greens called in English?
Likely brassica rapa perviridis.
It does not have a specific name in English, for the simple reason that it's not traditionally grown outside Asia.
If you find it labelled in English, it will be alternately "spinach", "mustard greens", "Japanese mustard spinach", or even "Japanese turnip greens", none of which are particularly accurate. Occasionally it gets mislabelled as "pak choi" here on the West Coast, as pak choi is a popular and closely related vegetable.
If a store is being accurate, it will be labelled "komatsuna".