Children are allowed on Ocenia cruises, but having recently been on one, there were no children on board, and certainly no facilities or activities for children. If you want to take children, then I think you would be better looking at other lines who do actively offer facilities and activities. My brother has cruised with his children several times on Royal Caribbean, where the activities were tailored to families, and had a fantastic time.
Yes children are allowed on the Oceana.
The OP asked about Oceania Cruises - not the P&O Oceana - so in reply - if you peruse the website you will find that there is no mention of children or of children's activities. I would think that the sort of person who books Oceania would be very upset to see hoards of children running around, and I think the child would be rather bored with no other children to play with. I would select another line.
Taken from FAQ on Oceania website:
Do you offer children’s programs / services onboard?
Although Oceania Cruises accepts Guests of all ages, the unregimented lifestyle and upscale ambiance onboard are designed for adults and do not provide for the care, entertainment, or supervision of children.