There's a lot of budget carriers out of EDI. In terms of mileage redemption, the majors are:

*A
EWR on UA
IST on TK
BRU on SN
FRA on LH
ARN on SK
YYZ on AC

OW
London on BA
BCN on VY via IB
DUB on EI via BA
SNN on EI via BA

SkyTeam
AMS on KL
LCY on WX via cityjet
CDG on KL

LHR on VS

I recommend BA for short intra European flights (Reward Flight Saver) and United for TATL to avoid fuel surcharges. And you might be able to get EI via BA but I was never able to.

Also keep in mind that the UK has a large air passenger duty on departures so it can actually be economical to fly to continental Europe, spend the weekend somewhere interesting, and then fly to the US.

Also, you should get a US AMEX now then do a global transfer to the UK. Recommend the 20,000 MR signup on The Preferred Rewards Gold card. Then once per year you can transfer to a US MR card at the currency conversion rate.