According to a post on a forum on Airliners.net, ANA should be taking delivery of 2 787s. ZA506 (LN 47, JA809A) will deliver tomorrow June 18 and ZA507 (LN48, JA810A) will deliver on June 19. Additionally ZA509 (LN56, JA812A) will deliver on June 28th according to the post. In terms of the Air India deliveries, that is still to be determined. It seems that the GoI has not given the go ahead as of yet on the deal between Air India and Boeing. If approval comes this week then I can see deliveries starting very soon thereafter.
The post also said that ZA100 (LN 7, JA803A) and ZA508 (LN 51, JA811A) will be delivered around July 31 and July 11 respectively and it may be possible for ZA135 (LN 66, JA813A) to be delivered during this month as well, this aircraft is still undergoing final assembly tasks on the apron. Qatar Airways, it seems may not take delivery this month but that Boeing will conduct its first flight and then send it to Farnborough under Boeing control and ownership after which Boeing will complete all necessary pre-delivery operations prior to handing the aircraft over to Qatar Airways. I also wouldn't be surprised to see further handover of aircraft to JAL and Air India in July.
Finally, I expect a line move late this week, around Friday the 22nd which should push out the first 787 for LAN, this is ZA536 (LN 68).