(Leaving aside the complication of FA Cup winners potentially also qualifying for Europe via other routes, like Chelsea will if they win the Champions League.)
Conflicting info from UEFA compared to the English FA, the BBC and other sources:
UEFA:
• [these teams] are joined in the play-off round by […] the domestic cup winners from the associations ranked 1 to 17 […]
• The victorious sides from [the play-off round] qualify for the UEFA Europa League group stage, and will be joined there by the ten losing sides from the UEFA Champions League play-off round.
No mention of cup winners going directly to the Europa League group stage.
The FAet al:
There are three places in the UEFA Europa League available: the FA Cup winners will enter at the Group Stage; the fifth place team in the Barclays Premier League will enter at the Play-Off stage; and the Football League Cup winners will enter at the Q3 stage.