I've had a good long hard think on this one. My thoughts are:

1. Agreed that the 'thing' is Voldemort.
2. The 'afterlife' is ever-present and you need to be in a state of 'death' to be witness there. This would explain why Harry can then talk to Dumbledore (already in the afterlife).
3. Given the Harry/Voldemort connection, the presence of Voldemort is 'awaiting' the final moment of Voldemorts destruction.

"I think" said Dumbledore, "that if you choose to return, there is a chance that he may be finished for good. I cannot promise it. but I know this, Harry, that you have less to fear from returning here than he does."

Harry glanced again at the raw-looking thing that trembled and choked in the shadow beneath the distant chair.

"Do not pity the dead, Harry".

Voldemort was in a state of death here.

I say this because in the next chapter:
'The death eaters had been huddled round Voldemort, who seemed to have fallen to the ground. Something had happened when he had hit Harry with the Killing Curse. Had Voldemort, too, collapsed? It seemed like it. And both of them had fallen briefly unconscious and both of them had now returned.

I think they were both in the afterlife while at Kings Cross Station. Voldemort - the hideous baby. Harry - the flawless one.

I think that when the final moment of death was to arrive for Harry and Voldemort - their 'afterlife' physicality is now pre-determined.