Beginning June 1, 2020, British Airways will fly direct to LHR 5 days a week, on a 787.
https://www.headforpoints.com/2019/12/18/british-airways-launches-flights-to-portland-oregon/
Introductory fares are good: $901 roundtrip for Basic Economy, or $1021 for standard Economy. Those are for June travel. Premium Economy looks to be around $1700.
There may be award seats available if you have miles on AS or AA to spend, but it looks like the awards have not been loaded into inventory yet.
.