To me, both sound natural. But I would say 'I've forgotten my keys.." Not a large difference, but that's just me.
But while 'I forgot my keys at home' sounds quite natural, I'd hear 'I've forgotten my keys' more frequently. Especially if it's in the morning and you're on the way out the door. Right after leaving the house, I don't think you'd say 'I forgot my keys at home'.
But again, that's just me.