Thought I'd give an updated solution for this, having just unlocked it on Rare Replay
This guide will focus on the quickest way to unlock the achievement, there's many Stop N Swap items out there. For more info and a full guide check out http://www.gamefaqs.com/xboxone/168561-rare-replay/faqs/5523...
All credit goes to the author Sakura Yule without his guide I couldn't have made this one. I've been told you need to have a Banjo Kazooie Nuts and Bolts saved game before the eggs will appear in the Banjo Kazzoie XBLA version, so make sure you do before continuing
To begin with load Banjo-Kazooie
and head into Treasure Trove Cove.
Once in the cove head to the sandcastle, with the castle behind you look out to sea.
You should see a box with an extra life floating out in the water near a cave.
Swim out to that cave and head on in.
Once inside head around the mountain to the spring pad, use to jump up and then climb/fly your way around the mountain to the top.
At the top you'll see a big egg, grab it and Banjo will talk about unlocking something in Banjo-Kazooie: Nuts & Bolts
Now load Banjo-Kazooie: Nuts & Bolts
Start a single player game and when it loads you should see a cut-scene.
Head to the location shown in the cut scene and grab the box.
Bring the box back to Mumbo and drop it on his delivery pad.
Speak to him and the achievement unlocks.
I made this video to help explain it all. Do Not Use The Sandcastle Cheats!
you can enter cheats in the sandcastle to find the Stop N Swap items, it turns off saving so you can't actually trigger the items in Banjo-Kazooie: Nuts & Bolts
Any questions, just ask.