Does he need a licence if the entire map is built programmatically and no TFL 'resource' (as in artwork) has been used directly?

Yes. I've spoken to TFL Licensing a few times about their map and basically they will claim copyright over any map that features the circle line in a "bottle" shape.

There are also restrictions over the font, logos and symbols.

I have seen people get around it by making their own version of the map. Those maps usually end up having the stations match their respective locations to each other more closely. They naturally don't look as 'good' as the official map though.

But where does the copyright infringement start? Could you technically get around it if you changed a couple of the colours and fonts around and used slightly different angles?

#whole-map { rotation:180deg; }

It might result in a few confused travellers, but hey.

Yes, otherwise it's copyright infringement. I've been working on a mobile app for some time. As an indie developer, the map license is almost fatal for my project.

I have however seen people get around this by making their own version of the map; often opting for a map that more closely resembles the true relative positions of the stations.

Naturally this CSS map is an almost 1-for-1 copy of the original. But how far would it have to diverge to not be copyright infringement? Different colours? Use rounder angles ?

Yes.