With the release of the Wayfinder's Guide to Eberron for 5E, can this book be used according to the "+1 rule" for characters in Adventurers League play? Can you play as a Warforged regardless of the adventure setting?
See The D&D Adventurers League Player & DM Pack which includes player's guides for the Forgotten Realms campaign (including the Season 8 Waterdeep Dragon Heist & Dungeon of the Mad Mage and Season 4's Ravenloft material, among others) and the stand-alone Eberron Campaign.
Each has a section on Character Creation which includes "Choose a Race and Class".
Your 1st-level character is created using any race and class options in the Player’s Handbook, Xanathar’s Guide to Everything, and the Wayfinder’s Guide to Eberron. Bugbears, goblins, and hobgoblins from Volo’s Guide to Monsters are also available for use. No other resources are available—even if other campaign documentation (i.e., certs) otherwise allow it.
The Forgotten Realms section is rather longer, but does not include the Wayfinder’s Guide to Eberron in the list.
It references the +1 rule you mentioned in the question:
Your 1st-level character is created using any race and class options in the Player’s Handbook plus one other resource from those listed below (the “PHB+1” rule).
This clearly indicates that the +1 rule means "PHB +1 from this list" and not "This list +1 from somewhere else".
(NB: The Realms guide has this rule, but there is no sign of it in the Eberron guide).
So you can play a Warforged in the Eberron setting, but not in the Forgotten Realms (plus Ravenloft) setting.