No. I think in a previous thread you mentioned that you had EXP status, which is a One World Emerald tier. 1W's published alliance-wide Emerald privileges include priority check-in, preferred boarding and seating, priority standby/waitlisting, and lounge access— but not free upgrades.
To get into Premium Economy, you'll most likely need to pay for it, either by upfaring your existing booking or paying at check-in where available. (BA may also sell upgrades at the gate or even on-board, but I'm not sure). There is a slim chance you might get moved up if the WT cabin is oversold, but you'll be far down the list.
First, a word on terminology. Products like AA's Main Cabin Extra, UA's Economy Plus, and Delta's Economy Comfort are more of a preferred seating scheme than true premium economy. They give you the identical seat and service level of any other economy seat, book into the same fare classes as regular economy, and are differentiated only in that you have several inches of additional legroom (and may board in a separate boarding group, since the seats are usually clustered together).
True Premium Economy such as on ANA, Air New Zealand, or Turkish is a separate cabin between regular economy and business, booking into higher fares and offering upgraded seats, IFE, and/or food/drink over regular economy. Merely having a certain credit card is enough to get you into United Economy Plus, but don't expect to get into EVA Elite Class without paying extra, any more than you would expect to get into business or first.
Second, remember that alliance privileges are a subset of airline privileges. Most airlines will treat their own frequent flyers differently from (better than) those of alliance partners; AA, for example, grants automatic status-based single class upgrades for Executive Platinum members on domestic flights, but does not extend that privilege to OneWorld Emeralds (e.g. EC Gold, Qantas Platinum). Why should BA give AA members more than AA gives BA members? For its part, even for operational upgrades, BA reportedly processes 1W Emeralds below EC Silvers.
Third, complimentary upgrades are a rarity on European and Asian carriers in general. They have a different business model for the premium cabins from the U.S.-based carriers, owing to differences in market structure and customer preferences, and simply giving away access is not in the offing.