Hi,
I fly through Oslo a few times per year.
On the OUTBOUND journey:
If you have an umbrella fold pushchair, you can check it in, take it to the gate and ask if you can bring it on the plane with you. They will often comply if you ask nicely.
It IS a long walk on the other side. And the luggage trollies with seats that aznerak mentions are normally not found where you disembark the plane in Oslo.
Try and get a seat so far at the front of the plane as possible, as the wait at passport control can be excrutiating, better get there quickly.
On your RETURN to the UK:
Check in your pushchair to its final destination. The trollies are found just after security.
I am not too familiar with Oslo. I havent been there with children for over 3 years. Family friendly restaurants? I did not actually find that many.... Most wont let you take your buggy in. If you see a restaurant/cafe/coffeeshop with lots of buggies parked outside you can be sure it is family friendly! Seriously. It means parents take their kids there!
Peppes Pizza does the best pizza in the world. They do childrens menues where the child gets a pizza, a juice, a balloon, and a dvd to take home. They often have play areas. They are quite "upmarket".
Akershus Festning, Vigelands Parken, The Edward Munch Museum, Aker Brygge (the pier),The Viking Museum, National Galleriet are places of interest.
And if everything else fails, feed your children waffles with jam.
Hopefully somebody else will be around to advice you further.
Have a great trip!