Never ordered a bike through the LBS because I've always bought used, but I think in large part that's dependent on supply from the MFG and out of their hands.
Parts -- My LBS does orders on Tuesday and will have the parts in by Thursday (with a couple of exceptions for certain boutique vendors.) So if you order something on Friday, you won't get it until the following Thursday, but if you order on Tuesday, you will also get it on that same Thursday.
1. For most items, when I factor in the cost of shipping, the LBS price is comparable.
2. When I do order online, I rarely order from Amazon because their prices and availability are rarely as good as Jenson, Niagra, ModernBike, etc.
3. Amazon Prime's 2-day shipping is often more than 2 days. It's usually only 2 days if you order by Tuesday; and sometimes they don't fill the order and ship it for a day or two after you order it, so I have had frequent instances where 2 = 5 in Amazon math.
4. Of course, buying certain parts from the UK saves a lot of money. But the trade-off is slow shipping. (Still waiting on some winter bibs I ordered 2+ weeks ago.)
In conclusion, I've found for that most of the time ordering from the LBS is roughly equal to ordering online in both terms of cost and delivery, with the exception of the UK vendors.
But my shop, I think, is better than some with orders. YMMV.