As far as I know, the AmigaOne isn't going to have this capability, either. (Basically, the AmigaOne is just PowerPC reference board with a custom UBoot -- it doesn't have any on-board video, and uses standard PCI/AGP cards, like you'd find in any PC.) Essentially, it would be the same as doing the editing on your regular PC. (Provided you could even find the software to do it on an A1.)
If you want to do analog-type video effects / live switching / etc, the Video Toaster is about the most common solution. This is why you still see them a lot in local cable studios. Especially ones that use older cameras, or do live broadcasts. They either have one of the Video Toaster cards for an original Amiga, or one of NewTek's newer PC/Windows-based VT cards.
But, if there isn't a specific reason you need analog video... I'd recommend just handling it all 100% digital on a newer PC or Macintosh. It's a hell of a lot easier (and cheaper) than messing around with time-base correcting various analog signals and such... Burning digital video straight to DVD gives a lot nicer end product than sending 15khz analog video to tape. (And if you need a tape, just make it from the DVD! ;-) )