I've worked in HR in the UK and we required a copy (that we made) of the BRP/visa. We usually took it sometime in the first week of the new hire, but we couldn't take a copy without viewing the original.
It's not likely to happen, but if the company is audited, they could get a fine if they don't have the immigration docs proving all their employees are legally allowed to work in the UK (BRP/visa + passport or just UK/EU passport for every employee).
Best to check with the HR department of your new job - they may be able to wait a few days after you start for you to find your BRP.
Today, I'd take everything you can relating to your BRP (card number, letter that came with the BRP and FLR/ILR approval letter if you've got it).
Good luck on your first day!