I expect that in the next 8 weeks, Boeing should be able to deliver 787s to Air India, JAL, ANA, Qatar Airways, ILFC (for Norwegian), LOT Polish Airlines, Hainan, Thomson Airways, and China Southern. Thus far, 787s for Hainan, Air India, ANA, Thomson, LOT Polish, and JAL have entered production test flights with the revised lithium ion battery set up and there is a pattern.
Boeing is conducting test flights roughly in order that each aircraft emerged from final assembly. There have been exceptions most notably a Hainan airframe which does not have any engines and a couple for LOT which are expected to be delivered later in the year. If the current pattern holds then I expect that from Everett the following will be making their first flights in this order:
ZA383 - China Southern
ZA318 - TUI
ZA319 - TUI
ZA513 - ANA
ZA185 - JAL
ZA465 - Qatar
From Charleston these aircraft will be making their first flights in this order:
ZA434 - Hainan
ZA243 - Air India
ZA384 - China Southern
ZA385 - China Southern
I estimate that Boeing can deliver between 12 to 14 787s by the end of the second quarter including a few to Chinese carriers who are expecting that the 787 will be finally given its airworthiness certificate by Chinese authorities next month.
Prior to the grounding it took Boeing about 6 weeks after rolling out a 787 to the flightline to conduct all the necessary company and customer tests, finalize paperwork and make final delivery. I see no reason why that would change now, though some aircraft that have already had quite a few test flights done prior to the grounding won't take that long since much of the flight testing is already completed.
Flight testing will continue to pick up pretty aggressively over the next few weeks as Boeing needs to work down the inventory of 37 787s that have built up over the last few weeks.