But Plato has Socrates speaking of souls going to 'heaven' or a 'hell'. Only the worst souls stay in hell, the rest are recycled. This is in the republic I think.

True, but it was Aristotle's thinking on the matter that formed the basis for most western speculation on the soul until the renaissance, not Plato's. In fact, between the end of antiquity and the fifteenth century, of all Plato's works, the Latin west only really had the Timaeus in common circulation (the Arabs had a bit more, including parts of the Republic, but they were also heavily Aristotelian in their thinking. The Byzantine world was perhaps the only place to have a significant amount of Platonic material to hand). The only significant Platonic thinking to reach the west was the solidly christianised Neoplatonism of Augustine.

Plato's position on the matter is harder to recover than Aristotle's - largely because the majority of his thought on the immortality of the soul (corporeality is not as a big concern for him as it is for Aristotle) is contained in the earlier socratic dialogues and put in the mouths of people who may or may not have other motives for their speculation. In the Apologia Socrates is made to say that he doesn't know what happens to the soul after death - and nor do any of his detractors - but in other dialogues he makes positive statements, or at least suggestions.