Your father may be subject to Capital Gains Tax on the disposal of the share gifted to you. More guidance can be found here:
Tax when you sell property
You usually pay Stamp Duty Land Tax (SDLT) on increasing portions of the property price when you buy residential property, for example a house or flat.
Stamp Duty Land Tax
If your father is the beneficial owner of the income from the property, then he will need to declare it on his Self Assessment tax return. You would not need to report income from this property as your father is the beneficial owner of the rental income.