If you are an employee, it is normally up to your employer to ensure that they have the information necessary to pay you on time (typically, this would include contacting you and/or others in the period after the timesheet deadline has elapsed and before the information is needed for payroll).
If you are not an employee but an independent contractor it is typically up to you to invoice companies for billable hours, to chase payment as one business to another.
It sounds like the 'hiring company' is an agency and there may be additional laws in your jurisdiction covering this relationship, but in the first instance it is up to them to pay you, irrespective of what they have or have not received from the client.
The legal recourse in each case will differ significantly, and although generally, absent evidence to the contrary you're normally an employee, you need to find out from your contract, your payslip, or from a qualified professional (union rep, solicitor, etc.) in your area which applies.
While you seek professional advice, I suggest promptly sending a dated letter (of which you keep a copy), explaining the amount you expected to be paid, the period of work the pay covers, the number of hours you're paid for (if this varies), and the date the payment was due.
Follow this up with a phone call. Make it clear that you got the paperwork on time and never heard anything from them to indicate they'd not received it. Don't bring your bills into it. You need enough cash to deal with a missed payment (including if you or they decide to end the relationship), and implying that you are living hand to mouth weakens your bargaining position.
Separately, you should also ask in writing for the signed copy of the terms of your employment (in future: never sign anything you don't get a copy of). You will at some stage want to move from this hiring company to someone who actually pays you on time and you will likely want to know whether you can work direct at your current placement. Obviously, don't tell the hiring company why you want this.
Finally, be on the lookout for other work. If you do accept another position, it's entirely appropriate to explain to your current placement that you are sorry to go but they need to find an agency that pays its staff on time.