I would just contact the league with the facts, so tell them that the blue manager believes the opponents played ringers, and that the player you cautioned didn't appear on the WGS list of names. Might also be worth providing a physical description of the player as that might help the league with any investigation they carry out. Beyond that it is out of your hands.
As an aside, it can be very difficult for grass roots leagues to bring a case on accusations of playing a ringer. If there are no ID cards and no video footage it is essentially one team's word against the other, and it isn't uncommon they will show the referee a photo and ask him or her if that was the player. Happened recently in the Womens FA Cup, tier 7 club Hammersmith beat tier 6 Bromley 4-0 but in doing so played a keeper that wasn't registered to them. This was quite an easy one to sort out though as she seemingly used to play for Bromley so they all recognised her and noticed the absence of her name on the team sheet. They've been kicked out and also banned from entering next season.