To what extent can it be said that the events detailed in "The Monkey’s Paw" were supernatural in nature?

There is no way of knowing whether any of the events detailed in "The Monkey's Paw" were supernatural in nature. The author W. w. Jacobs intentionally cautions the reader that all of them could have been coincidences. In the opening of Part II he has Mr. White tell his son:

"Morris said the things happened so naturally," said' his father, "that you might if you so wished attribute it to coincidence."

White's first wish is modest. He only asks for two hundred pounds. That happens to be the amount the company pays for the loss of his son Herbert, but it seems like a reasonable sum in those days, since the company does not take responsibility for the accident. It also seems reasonable that Herbert should have gotten into an accident on that particular day because he stayed up late talking to their visitor and drank more whiskey than usual. He could have been tired and less attentive.

Then when Mrs. White asks her husband to use the paw to wish for Herbert to come back to them, Mr. White insists that the granting of the first wish was only a coincidence.

"We had the first wish granted," said the old woman, feverishly; "why not the second?"

"A coincidence," stammered the old man.

Mr. White only consents to make the second, truly diabolical, wish because he does not believe in the supernatural power of the paw. And here the author deliberately reminds the reader that all this could be the results of coincidence. 

Late that night when someone begins knocking at the door, it could be Herbert. On the other hand, it would be some lost stranger who is only seeking directions. The author establishes at the beginning that this is a dark, out-of-the-way, barely inhabited suburb.

"That's the worst of living so far out," bawled Mr. White, with sudden and unlooked-for violence; "of all the beastly, slushy, out-of-the-way places to live in, this is the worst. Pathway's a bog, and the road's a torrent. I don't know what people are thinking about. I suppose because only two houses in the road are let, they think it doesn't matter."

There are only two occupied houses, and the hypothetical lost stranger might have already tried the other house. He would know there were people at home at the Whites because he would have seen the lighted window.

He sat until he was chilled with the cold, glancing occasionally at the figure of the old woman peering through the window. The candle-end, which had burned below the rim of the china candlestick, was throwing pulsating shadows on the ceiling and walls, until, with a flicker larger than the rest, it expired. 

If it was a lost stranger, he might have kept knocking louder and louder because he was desperate. There was no one else to turn to. And when the knocking ceases after Mr. White makes his third wish, it could only have meant that the stranger had at last given up. On the other hand, of course, it could have been the horribly mutilated and decayed Herbert returned from the dead to move back in with his parents. The reader will never know--and may never want to know. The reader might feel fortunate in having been spared the necessity of seeing what was standing outside that front door!