Hi, I just saw this thread pop up again. I don't know if anything has changed in the last 2 years, but all we had to do was apply and pay the passport fee of £46 for our 2 kids (plus whatever the courier fee was). It was really easy and we had their passports within 2 - 3 weeks. Like I said, I don't know if anything has changed, but there definately was no registration fee of £749!
That would be because your childrens' situation is actually quite different to Cloverbea's.
As you were born and raised in the UK (UK citizen otherwise than by descent), your children are UK citizens by descent and have been UK citizens since the day they were born. Because of this they did not need to register as citizens - all they needed to do was apply for UK passports.
Cloverbea's children are NOT currently UK citizens, because their father was born in the US and is a citizen by descent (like your children are). He received UK citizenship through his mother, but this citizenship does not automatically pass to his children.
Normally, Cloverbea's children would need visas to move to the UK and would not be eligible for UK citizenship from their father at all. However, the only reason Cloverbea's children may be eligible for UK citizenship is because their father lived in the UK for more than 3 years when he was younger and they can qualify under that criteria.
So, in order for them to gain UK citizenship, they will need to first register as UK citizens and pay a fee of £749 each and then once their registration has been approved, they can apply for UK passports from the US (which are now £53 plus £19.86 courier fee).
Essentially, Cloverbea's children are in the same situation as your wife, in that they aren't automatic UK citizens, but may be able to register as British.