BEP make some very valuable and additional points here. 99% of contractors/interims work through either their own limited company or an umbrella company, either of which then has a contract with the agency who then have a contract with the client.
The value in an umbrella company is that you don’t have the (minimal) hassle and expense of running your own company (annual returns, VAT, compliance, etc) - in essence they let you borrow theirs. The problem is that there have been some spectacular implosions of umbrella companies, and a few of the smaller agencies, where they go bust owing the contractor money. Under such circumstances you will be reliant on the appointed liquidator to send you on about 10p in the £ owed, probably in about a years time! The end client has NO obligation or likelihood of paying you owed moneys as they have almost definitely already paid the umbrella/agency with whom they have a contract.
All reputable agencies will expect you to submit a client-authorised timesheet and invoice by around the 5th of every month (for the preceding month) and in return will pay you the full invoice amount by the15th -20th of the month. They then get the money back from the client. It is ILLEGAL for an employment agency to not pay their agency workers because the end-client has not yet paid them. Many, many contractors exempt (by signing an opt-out letter) themselves from that law in order to improve their IR35 position, but nonetheless, all reputable agencies voluntarily adhere to that principle as general market practice.
If you ever find yourself needing to nominate an agency (your client does not have an approved one) and you don’t know a reputable one, then let me know – I can happily recommend some of the larger agencies that have always paid me on time.
Also, if you DO get your contract direct, and then need to nominate an agency for invoice/payroll purposes, then make sure you get them to delete their standard terms relating to their "protection of interests" - this is the section that says that if you work through them at the client, you can then ONLY work through THEM at that client for a stated period time (often up to 6 months after the contract ends). This is only relevant if they have invested time in finding you the job in the first place.