No matter where you are joining, never serve your notice/resignation until you have the new offer letter, in writing, signed by you, counter-signed by the company, and all conditions on the offer are formally lifted (i.e. criminal background check, university degree validation, etc.) and the new company has emailed you confirming they've lifted the conditions and the offer is now unconditional. Anything less than this, and you're inviting a world of hurt (i.e. laid off on first day of work, offer rescinded right after you quit your current company, etc.).
1 month with no follow-up? Sounds super sketch for any reputable Fortune 500 company or equivalent. Reach out to your recruiter.