My monthly immediate pension point (IPP) payment date is the date that I left Regular service, my lump sum rocked up about a week after I left - that was 11 years ago. It has no connection to my joining date. My index linking started on my 55th birthday and now my pension is uplifted every April. Again, no connection to leaving and joining dates. Payment times are all connected to “pay runs” on JPA.
For BV be prepared for getting your final month of pay in arrears first, then back dated pension and monthly pension in a backdated blob. The lump is normally paid separately in my experience. I suspect part of your delay will be the new AFPS15 scheme due to the career averaging element needing to be calculated after you have been paid your final monthly salary - that is likely why they state 30 days for payment. 36 days is a bit on the long side though, but I thought you said you were on loan service which may have caused a bit of extra added complication?