Given the delays in the system I suspect there is a very little chance of you getting your sons passport in time, you say the documents were received on the 20th April is this via the post tracking or has the application updated for documents received? When did you make the application?
All applications for a 1st adult passport have to have an interview, so there is a chance your son will have to be interviewed – it’s not a light in the face style interview, more an informal chat, just to make sure they are who they say they are.
Given the time until you travel I would contact the passport office to see what your options are, but be prepared for some bad news or to pay a lot more money to speed things up – but reading posts elsewhere even this route no longer seems to be a guarantee your passport will arrive in time.
Good luck!