HI, yes I did the same thing , I waited till I became British before we applied for my daughter's Indefinite Leave to Enter.
We did accomplished vaf4a family settlement application and we registered an account here :
https://www.visa4uk.fco.gov.uk/home/welcome and filled in as much information as we could. I think its worth applying for the premium service as they were very polite and helpful.
I did authorized my mum to apply for my daughter's passport beforehand, we sent a letter of authorization which we got notarized at the Philippine Embassy in London and send it to my Mum through DHL.
Health surcharge , is not included with our application for ILE in 2014 but we did accomplished Appendix 1 to go with the settlement application.
Yes, CEO sticker is required in your daughter's passport , we applied for it early in the morning and we get it the same day.
TB Clearance Certficate is also needed in the application (my daughter has been visually check by a lady GP didnt undergo any scans and xrays) and we got the clearance certificate the same day but we booked it with IOM before it's due to submit documents with VFS.
International Organization For Migration/
Address: 8741 Paseo de Roxas, Makati, 1200 Metro Manila, Philippines
Phone:+63 2 848 1260
Timeline for my daughter's ILE: six weeks and that's with my mum's visit visa at the same time.
My other advise is to check the latest news and updates at. gov.uk website because as you know the rules and visa fees seems to be continuously changing.
Good luck with the application , hope it is successful.