I don’t think Rey is a Skywalker — I’ll be seriously disappointed if she is.
My take on her force powers developing quickly:
She ‘is’ a force-sensitive person — perhaps in the days before the Empire, she would have been trained as a Jedi from childhood.
When she touches Anakin’s/Luke’s lightsaber, she receives visions through the force. It has been established before that the crystals in lightsabers can syphon off excess force-powers from those who carry them (KoTOR II). Consider; this is THE lightsaber that Anakin fell to the dark side with — he cut off Windu’s hand, allowing Palpatine to kill Windu. He then sacked the Jedi temple, killing dozens of Jedi, including the younglings. He then travelled to Mustafa and killed the seperatist leaders. If lightsaber crystals do absorb some force power from their users, then surely this lightsaber is highly charged with energy. This is why, when Kylo Ren sees Finn with the saber, he demands it — not because it was Luke’s, but because it was Darth Vader’s; the one he fell with. Surely, he sees it as a powerful artefact of the dark side. As he hero-worships his grand-father, you can see why he would consider this ‘his’ property.
Going back to Rey, as a force sensitive, I think she may have gotten some sort of ‘jump start’ to her own powers (why this didn’t happen to Luke in ANH, I don’t know) from touching the saber, as it is only after this point that she really starts to develop any kind of control over her abilities. Perhaps, the visions she saw also transferred some memories from Anakin or Luke to her subconscious, which allowed her to learn to control her powers at an accelerated rate?
Obviously, I don’t know for certain. But, for me, this comfortably explains ‘how’ Rey develops her powers so quickly (at least, until Episode VIII comes out!)