Thanks for the tips everyone. Yes, indeed coming down from Glasgow. Lesson learned, land at LHR before a LCYCW trip. I will likely take the DRL to Canary Wharf as I have not been there before in my many trips to London. I will also check out the Emirates webpage linked above and maybe take that option if I am full up from the BA breakfast on the way down.
Seems like LCY needs a nice restaurant like Plane Food. I almost didn't mind not have lounge access when I was building status.