Is there a word for the flavour shared by onion, spring onion, shallot, leek, and chive?
Conversationally, "oniony". Everyone will understand that, and tend to say it naturally.

In a more serious cooking context, you could also get away with "allium flavor", though likely not in everyday conversation. There are a lot of alliums, including onions, garlic, scallions, shallots, leeks, and chives, so if you refer to the whole group, you're pretty obviously referring to the overarching common flavor. For example, On Food and Cooking has subsections called "The Flavors and Sting of Raw Alliums" and "The Flavors of Cooked Alliums".