I flew on Continental through Newark years ago and remember seeing a flight for Edinburgh boarding while I was waiting for my own connecting flight. Continental was sucked into United, so I guess it makes sense that it's United listed with direct flights from EWR to EDI.
Routes can change, but currently according to the web sites for GLA and EDI:
http://www.glasgowairport.com/destination-inspiration/route-map/ .. something I've not heard of, Atlas Air, is listed as going direct to GLA from BOS and also BWI; United and Jet2.com (I don't know it) are direct from EWR; and US Airways is direct from Philly.
http://www.edinburghairport.com/flights/destinations-and-airlines .. lists United for EWR and US Airways for Philly.
Atlas Air (according to web searches) is some kind of charter co. Jet2.com's web site doesn't encourage my confidence, so United or US Air are likely options.
I envy you. A direct flight is a lot less stress on the body, so you've lower traveling fatigue.