docwood,
Nassau to Egg is certainly one way - there is a decent, but small anchorage just to the east in Royal Island. We've come down from Little Harbor in the Abacos to Royal a couple of times, then shot straight down to the top of the Exumas from there. We were in a cat, drawing 4 ft.
If you wanted to do Eluthera in a south to north trip, you could cross from Nassau SE to Highborne Cay for a stop, then across to Powell Point at the southern inside of Eluthera. From there you can dribble up the inside
parts of the island with good wave protection from the east. We anchored in Governor's Harbour - small town with a grocery and some nice architecture in the homes on the hills. One interesting anchorage is in Hatchet Bay Harbor - narrow entrance that opens up to a protected bay.
Current Cut, your
passage to the north part of the island, has a pretty strong flow, so make sure you time that correctly to take best advantage.
From there we only ventured to and through Spanish Wells on our way back to
Abaco. No experience with the lower Harbor Island question but the
charts look doable on a high tide, should you venture all the way around the eastern side to enter by Whale Point. Best of luck - enjoy your cruising and Pecan Pie! Pat
PS I have thrwon out jumping from mid Exumas over to southern Eleuthrea as 
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