The company was Alldays & Onions, but the cars were just named Alldays so far as I remember from studying cars of that period.
I've just had a look in my Beaulieu Encyclopedia, and you're absolutely right. I had never realised that - certainly the NMM have a couple, and they're displayed as Alldays & Onions. This is a turn up for the book! We already have 2 cars of that marque on the AP search, so I've decided to leave it as both names in order to avoid confusion, although if AP is to be considered an online source of automotive fact, perhaps I shouldn't. Oh decisons, decisions......
