I vaguely remember having to call back after being misinformed in the past. I've also used BA's You First email. In my experience phoning works better. Typically, after calling I receive an email acknowledging my request:

"As we discussed earlier this evening, please find attached a copy of the onboard food menu and wine list for your upcoming flight to ...".

As you mentioned, Soupey202 the menu for May should indeed be available to you now.