Short answer: Grant: up to 20 working days (pre-contract) or 50 working days (post-contract). Loan: 1-2 weeks.
The First Home Grant takes up to 20 working days if you apply before signing a contract. The First Home Loan pre-approval takes 1-2 weeks from your bank. If you apply for the grant after signing, it can take up to 50 working days. Apply early. Do everything in parallel, not one after another.
Pre Contract: 20 working days
Post Contract: 50 working days
Loan Pre Approval: 7-14 days
Total Parallel: 3-4 weeks if done together