Just a couple of thoughts which I hope will help you find a solution.

I am pretty sure that you can only pick up an item deposited at a lost luggage service with the original receipt/ticket they give you at deposit, so you would need to get that off your sister to be able to pick up the backpack. I assume she is coming through Gatwick given that is the location you mention for lost luggage, so is there any chance of simply meeting up with her and picking it up?

Not sure if the backpack will have possessions in it? I assume so if the idea is for your sister to check it in/take it with her on the flight over from Jersey. If it doesn't have items in it could it be an option to post it over? Excess Baggage Company at Gatwick quote £7.50 for first 3 hours, £12.50 for up to 24 hours, etc. so worth checking how much you may be in for.