Thank you for your suggestion. My employer has already used that service for me but they said it'll take up to 10 working days. So i was just wondering if I could get the email/confirmation letter from Home Office a bit earlier than that.

Hi, I've made an application for a replacement card (indefinite leave to remain) in June. I've had my first day at my new job yesterday. Unfortunately, I couldn't continue to work as I didn't have anything to prove that I have the right to work in the UK apart from enrolment letter from Home Office....