According to Rutherford, electrons orbit around nucleus just like planets. Since they are revolving, their motion is accelerated. When a charge particle (electron) accelerates it give off radiation and looses it energy. So the electron of Rutherford's model will continuously radiate- loose energy and spiral towards center but no external force is being exerted on the atom so the angular momentum should be conserved of the system. So when the electron initially starts to spiral inwards, its velocity will increase to compensate for the reduced radius but when finally it hits the nucleus it will have zero velocity. And angular momentum of the nucleus remains zero the whole time. So how will angular momentum remain conserved when it hits the nucleus?
I think the Rutherford model should be completely consistent, being defined in terms of classical electrodynamics. That it doesn't correctly describe the atom is beside the point.
In this classical electrodynamical model there shouldn't be any difference between electrons around a nucleus and a light ball around a heavy ball (or other shapes you want to give them), as long as the masses and charges are scaled accordingly, and also the radius if you want to know exactly when the collision takes place. Note that if they were both points, the distance would get ever smaller but never 0 and there would be no collapse.
What would happen exactly depends on what happens when the electron hits the nucleus, something that is not part of the model. If the surface of the nucleus acts as an impenetrable and frictionless barrier, it would start sliding or bouncing around it. If on the other hand the two would somehow fuse into something new, the combined new object would be spinning.
EDIT As AnnaV and Emilio Pisanty remarked, the radiation will carry away part of the angular momentum, and it is not true that the electron-nucleus system would have conserved angular momentum. It would never lose all of it though.
For the model you're describing, the angular momentum of the system is conserved in the same way that energy is: it isn't.
More precisely, neither energy or angular momentum is conserved for the proton+electron system by itself, because it is not an isolated system, as it is in contact with the radiation field. However, once you include the energy and angular momentum content of the radiation, then you do get an isolated system, and both quantities are conserved.
It's important to emphasise that there absolutely needs to be angular momentum radiated away - the orbit cannot decay without it. Circular orbits have minimal energy for their angular momentum (and maximal angular momentum for their energy). If you want to decrease the energy of the orbit by radiating away, then you also need to get rid of angular momentum. This is also radiated away, and becomes part of the angular momentum content of the radiated fields.
At the endpoint of the spiral, you're strolling right out of the physics postulated by Rutherford, so what happens next is basically "in which way do you want to extend the model?" - and the answer will depend on the details of your choices when building that extension. But if you want a model where the nucleus has finite size and the electron merges into it, from a tightly knit high-velocity circular orbit, then the nucleus will need to start spinning, and that system will have a conserved total angular momentum within that interaction.
Finally, a word about the epistemology here. As has been pointed out, the Rutherford model is wrong. At best, it is a stepping stone to the Bohr model, which is less wrong (but still wrong); the dynamics of electrons in atoms are governed by quantum mechanics, period. The Rutherford model is encased within classical electrodynamics, and as such it does not describe reality. However, precisely because it is encased within classical electrodynamics, it is possible (it has to be possible) to provide a full analysis of the configuration within that classical electrodynamical framework, which explains what the model predicts will happen (even if it is inconsistent with reality, as in, say, an electron inspiral towards the nucleus) while fully respecting the internal rules of the formalism (including, in particular, the conservation of angular momentum).
the important point here, it seems to me, is that rutherford's model was incorrect- inasmuch as it predicted effects which were not observed, as you today and others back in the day pointed out. Because of this, there isn't any fundamental way to make sense of it, as you have discovered.