Hi all - thanks for your replies!
Ceremony was yesterday (now a citizen - yay!) so I'm just coming back to close-the-loop on this question in case anyone has the same question in the future.
Seating was mixed, with guests and applicants all sitting together. When it was time for the pledge, all the applicants just stood up. The kids came up to the front with us to collect the certificate and have our photo taken - again, absolutely no issue.
Having the kids there sitting with us was absolutely no problem - there were heaps of people there in the same position. The speeches weren't too long, but there was a fair bit of waiting around as all the applicants were called up to the front to collect their certificate and get a photo with the Mayor.
PLOT TWIST: I have to admit to being slightly dishonest in by opening post - when I phoned the local council office to ask about this, they didn't just repeat what was on the website.... they actually told me that the kids (1 and 3) would not be allowed to sit with us during the ceremony and that we would have to either leave them at home with a babysitter, or bring a guest to the ceremony who would have to sit with them in the "guest area". I'm glad this turned out to be total BS (as I suspected at the time) - but thanks again for your real-word experiences, on the basis of which we just decided to ignore what they said and turn up with the kids in tow. As it turns out - this forum is a more reliable source than the lady at the council office!
Thanks again all!