Since 1937, this has been governed by the Regency Act 1937, which sets out the following steps to determine who would serve as Regent:
If a Regency becomes necessary under this Act, the Regent shall be that person who, excluding any persons disqualified under this section, is next in the line of succession to the Crown.
A person shall be disqualified from becoming or being Regent, if he is not a British subject of full age and domiciled in some part of the United Kingdom, or is a person who would, under section two of the Act of Settlement, be incapable of inheriting, possessing, and enjoying the Crown, or is a person disqualified from succeeding to the Crown by virtue of section 3(3) of the Succession to the Crown Act 2013; and section three of the Act of Settlement shall apply in the case of a Regent as it applies in the case of a Sovereign.
Prince William's other children are, of course, also too young to serve as Regent, so we pass down the line of succession to Prince Harry. Harry is not currently domiciled in the UK, so is not eligible. His children are also disqualified by both residency and age, so the Regency would pass to Prince Andrew.
Parliament could, of course, decide otherwise by passing primary legislation. This was the common practice prior to 1937; see, for example, the Regency Act of 1811.
For completeness, the other disqualifications listed in the Regency Act refer to someone within the first six people in line to the throne having married without the consent of the Monarch (Succession to the Crown Act) or are Catholic (Act of Settlement).