Note that BA have a strange rule about this, plus also details for Qantas.
This is taken from AskBA (answer no. 795).
If you are travelling with an infant who reaches the age of two on a date during your journey (i.e. departs under the age of 2 but returns over the age of 2), a separate seat is required for all sectors on and after their birthday.
British Airways will be pleased to ensure that your reservation reflects a booked seat for that portion of your journey.
The fare charged will remain at the infant fare for the entire journey, regardless of the fact that a seat has been booked for some sectors.
If you would like to ensure that your child has a seat after their 2nd birthday, please contact your local British Airways office from the contact details found if you click here.
This policy does not apply for passengers travelling on Qantas flights, on which a child turning 2 years old during travel must either continue to travel sitting on their parent's lap or pay a child fare for their remaining flights after their 2nd birthday.
Please note: Bookings should be made according to the age of the infant/child at the time of travel, not at the time of booking.
Therefore if the infant will be two at the time of travel for the start of the journey then a booking for a child should be made
Not sure how this would mesh with an OWE ticket but might be worth having it issued on 125 stock....