Statutorily speaking, K-1s do not have to be delivered to the partners until the 1065 and K-1s are filed with the IRS (the latest date being the due date of the 1065 including extensions). See §6031(b) and Treasury Regulation §1.6031(b)-1T(a)(1). I believe the penalty is $50 per K-1.
From the partnership's perspective, the partnership agreement generally speaks to when K-1s are to be delivered to the partners. However, there is usually a clause that gives the GP or Managing PArtner leeway in extending the due date. Additionally, many partnership agreements are silent on this issue. Practically speaking two general scenarios take place:
1) Schedules K-1 are sent to the partners on the same date the 1065 and K-1s are filed with the IRS; or
2) Schedules K-1 are sent to the partners before the 1065 is filed. The 1065 is then prepared based on the calculations that went into the K-1s. If any information changes, the partnership "trues-up" the numbers on the following year's 1065. Although not the technically correct way to do it (amending the 1065 and K-1s would be the correct way), it is the pragmatic approach that many large partnerships take because they are not audited every year by the IRS.
If you are adamnant about filing now you file now with the estimated information. You would need to include [|Form 8082], Notice of Inconsistent Treatment, showing "0" for the amount reported by the partnership on K-1 (because they have not been filed yet) and using the estimated amounts.
Once the finalized K-1 is received an amended 1040 would have to be filed. You would have to understand that this would increase your fees owed to your tax preparer. However, since it seems you want to get this finished, I don't think this gets you where you want to be since the amended return will still be hanging out there waiting for the finalized K-1s.
I believe March 15 is a deadline only for partnerships that have elected to be "electing large partnerships" under §775. See §6031(b).