Indeed the A12 is just an A road. It would have to have a specific order and signs banning you from it. The only A roads I see banning cyclists and pedestrians are ones about to go into tunnels etc.
Just after you come off the Redbridge roundabout going S on the A12 pedestrians and cyclists are prohibited. I don't know how long the restriction lasts. I suspect it may be all the way down to the river.
Google streetview link
Plus bikes aren't allowed in the Blackwall tunnel, as you mention, but I don't think the road is still called A12 at that point.
Come to think of it I don't recall any "no cycles" signs on the Blackwall Tunnel approach, so perhaps they are already restricted by then (possibly the restriction in the picture above is still in force)
I'm quite good at A12 signage trivia having driven it a zillion times. For example, did you know there's a
minimum speed limit of 10mph on the Blackwall Tunnel approach? I often find myself inching along through that zone and fear instant arrest for breaking the limit.
I can't recall ever seeing a cyclist on the stretch of the A12 between the tunnel and Redbridge in either direction.