You are correct in your assertion that pairs of charged point particles can interact magnetically in ways that seemingly violate Newton's 3rd law, and therefore also seem to violate the conservation of both linear and angular momentum. This is a fundamental result and it is the decisive (thought) experiment which forces us to change our viewpoint on electrodynamics from something like
charged particles interact with each other
to a field-based one that says
charged particles interact with the electromagnetic field.
What this means, and the key point here, is that
- the electromagnetic field should be considered as a dynamical entity of its own, on par with material particles, and it can hold energy, momentum, and angular momentum of its own.
The linear and angular momentum of the complete dynamical system, which includes the particles and the field, is indeed conserved. This means that in a situation like your diagram, where there is a net force and torque on the mechanical side of the system (i.e. the particles), there are corresponding and opposite net forces and torques on the electromagnetic field.
So, how much linear and angular momentum are there? This is a solid piece of classical electrodynamics: these momenta are 'stored' throughout space, with densities
$$
\mathbf g =\epsilon_0 \mathbf E\times\mathbf B
$$
and
$$
\mathbf j =\epsilon_0\mathbf r\times\left( \mathbf E\times\mathbf B\right),
$$
respectively. Once you account for these, it follows from Maxwell's equations and the Lorentz force that, for an isolated system, the total momenta are conserved. The details of the calculation are a bit messy, and so are the actual conservation laws; I gave a nice derivation of the linear momentum one in this answer.
Not all forces are central. For example the Lorentz force is not a central force.
However I suspect most of us would regard the distinction between the weak and strong versions of the third law as rather pointless. The third law is a statement that momentum is conserved, which is itself the result of a fundamental symmetry. Whether the force is central or not makes no difference to this fundamental principle. I would file this one in the some physicists have too much free time category.
Best Answer
Let charge A be at the origin, moving to the right (along the positive x axis). Let charge B be at coordinates (1,0), moving in the positive y direction.
A's magnetic force on B vanishes, since by symmetry the magnetic field due to A is zero at B's position.
B's magnetic force on A doesn't vanish.
In magnetostatics there can't be any radiation. If there's no radiation, then mechanical momentum is the only form of momentum we have. If Newton's third law fails, then mechanical momentum isn't conserved. This would lead to a violation of conservation of momentum, which is impossible. So no, the third law can't fail in magnetostatics.
It would have to be an experiment in which a large amount of momentum was carried away by radiation. Seems tough to me. Even if you build a very powerful and directional radio transmitter, the amount of momentum carried away is tiny in mechanical terms.