It may be worth considering using the services of an intermediary service to handle the reporting of timesheets and also the payment.
I have previously used a site called Upwork (it used to be called "Elance"), and while there are fees attached I found the reporting of hours to be very easy, and the client would automatically pay and there was a dispute process available should there be any concerns (not that I ever had to use it). Plus, they automatically paid weekly either directly into a UK bank or also PayPal. As this is a dedicated job, your potential employer could create an Invite-only job on here and send you an invite, and then you can negotiate terms. It's worth mentioning some of the fees can be quite high at first (20% cut of first $500USD earned, for example, before it drops to 10%), but on the whole I actually found the service pretty useful.
Bear in mind any income earned should be reported to HMRC. You should consider either, (a) setting up a Ltd Company and having payments going to that, and then paying wages/a dividend to you as a director - you will likely need an accountant to help with that. Don't just assume that you can work full-time and avoid any tax liabilities - and factor this into your daily rate so that you are not out of pocket. or, (b), make use of an Umbrella Company to handle the invoicing side, you would send them your timesheets, and they would then invoice the company for your time and handle the USD to GBP transaction.