In his booklet "Bickell of Milton Abbot", Alan Taylor states that David Bickell was a witness to the marriage of Joseph Veale in1802 and speculates that this could be Elizabeth's brother. I imagine you will know that the first (?) child of David and Elizabeth was named Sampson Veale Bickell.
I wonder if trying to trace a Joseph or Sampson might give you some pointers to the possible origins of Elizabeth.
I see Family Search Pilot has the baptism of an Elizabeth Veal on 5 June 1770 at Holbeton, daughter of Sampson and Margaret, but that does not seem very convincing to me. I cannot find a Joseph, son of the same couple. Apparently Elizabeth was living in Tavistock at the time of her marriage to David.