Your link shows the Dartford Crossing, an M25 bridge miles downstream of the City. The Blackwall Tunnel runs under the Thames at Greenwich and afaik just has the old school hanging metal blocks at height https://maps.app.goo.gl/N5xSF148ggLVTDtS8

It doesn't surprise me too much that police are on standby, a closure of either tunnel or bridge has a major effect on traffic all over London

There are additional traffic lights on the blackwall tunnel further in and a slip road out that can be used for overheight vehicles. I do remember having a 10-15 minute wait once while they sorted things out when a lorry driver got caught.

I'd have a feeling there are automated signs prior to the tunnel (or at least used to be) but I've not been through the tunnel for a year or so and things will have changed with the Silvertown tunnel opening.

I have seen someone not paying attention at the Rotherhithe tunnel and the roof of their van was a mess (and they're going to pick up a fine probably due to restrictions, the 2 tonnes gross weight limit is lower than a lot of van drivers expect)

Edit there were: https://maps.app.goo.gl/MP7fkhS394DJPQaZ9 If you zoom in you can see the overheight vehicle warning.

You're right of course.

Blackwall seems also to have two sets of lights and barriers, and an off-ramp in between. That's probably for fire safety too to close and evacuate the tunnel and get the emergency services in, but I imagine it's used for height detection too if a loud CLUNK on your truck cabin isn't enough.

As an aside, the person who signed the original heights as (13ft)(4m)(9ft)(2.8m) needs to learn a bit about UI design. Yes, two lanes, but the gap between the central two signs is far smaller than to the other sign for the same lane. Also 4m is just over 13 ft 1 inch, which there'd be space to include as there's already a 0 on the leftmost sign (and from the rightmost we see that decimals are allowed on signs). Guess we're going to rely on the CLUNK after all.