Why there's no relief
Relief needs you to live there as your main home. A first purchase that you let out pays standard rates.
When higher rates apply
If, on completion, you (or your spouse or civil partner) will own another home anywhere, the 5% higher rates apply. A buy-to-let through a company always pays the higher rates.
Example
£250,000 first property, let out, no other home: £2,500 at standard rates, compared with £0 for a first-time buyer living there.
Moving in later
Buying to let and moving in later doesn't bring back relief; it's decided on completion day.