I finally manage to post an answer but in the meantime you received many answers. Some of my points was described already, but maybe you will find something new and worthy :)
First of all, you are responsible for your part of the SLA. If means you should buffer all the problems and stop the domino effect, shielding the client from it. If you can't get the subcontractor make work done, then only way left is to find internal or external team to overtake the job.
Describe the situation to the subcontractor as well as client's impatience. He may do his work and have your active support with talks with the vendor or he may choose not to cooperate and tighten "the spiral". If I understand properly, first case brings him big chances of being payed (late) while the latter means he will only get additional problems.
Sweeten your relation with the client. Try referring to overall cooperation within the whole SLA or even offer something in exchange for extending the deadlines (additional support or extended period of of it). That way, you will buy some time to solve the problem.
Tighten the screw. Your original agreement with subcontractor may not include forfeits for delays but it doesn't mean you can not introduce deadlines. There are usually many informal aspects and "buttons" one can push. Put on the table as much as you believe is needed: subcontractor's reputation, your support/opposition, future engagements with this subcontractor (at least you have recommendation/criticism ability), agreement re-negotiations or formal actions that can be taken. Start it gently, not all guns blazing, and fluctuate slowly to the carrot&stick situation if needed.