We did OV2 in July with a 2 year old and it worked fine. None of us wanted to use the pullman berths so we took his 'Bob the Builder' ready bed with us and it worked a treat as there was plenty of room in the cabin. Don't worry about packing too many clothes given they have a self service laundry (£5 for a large load). The base camp was great as it had a family area at the back with a small paddling pool and loads of toys. My little boy didn't want to join in the sessions but the staff were quite happy for us to go to base camp and use the toys and encouraged us both to take part in the group sessions. OV make travelling with children very easy. There is a children's tea at 5.15pm, although my son prferred to eat later with us. We have just returned from another cruise and found the easiest way to wash our son was to just sponge him down in the shower tray - he refused to go in the shower with it on full blast.
Sounds like at least one of you will be up on a bunk! The two year old in a cot? pre book the cot! As you know these are roomy cabins with lots of wardrobe and drawer space. The tightest thing is the bathroom and bathing a small child in a shower is difficult and potentially dangerous so if you can try it out at home, do. I have found putting a towel down on the floor of the shower helps prevent slipping. A good idea is also to put away the kettle as it is too easy for small children to get at.
Your main problem will be the air travel and time difference upsetting their routine. Very young children suffer quite badly (far more than adults) from the change in pressure. So at take off and landing have them suck on a bottle or lollypop as they need to suck and swallow. Too many people believe children are frightened when really they are in pain. Moniter their behaviour for ear pulling, a sure sign of pain in the ear. Also they get very dehydrated, so compensate with water drunk little and often (good for you too). This will ensure they will feel well and have no problems in the toilet departement! You can get a "child aviation restraint" (put into google) and dont choose bulkhead seats as the arms do not lift up so the child can not lie down and put their head on your lap. Take books and toys (new!, baby wipes, couple of plastic bags for rubbish and nappies, and a change of clothes for them and top for you (in case!)milk,treats etc. Pre book child meals. OV is a synch after that. Common sense and be alert to obvious dangers particularly from other people not noticing that a small person is there, then walking/running through them knocking them flying! Seen that too often. Although there is a medical centre the cost should ensure your extra vigilance. Have a lovely time as there are great facilities for both you and the kids.