Child born in UK applying to register

So my daughter was born in the UK when i was on Tier 2. In a couple of weeks im applying for ILR and if all goes well should receive the BRP a few days after that. I would like to ask:-

1. Is it ok to register my daughter straight after I have recieved the BRP or I need to wait for a little while?

2. What is the shortest route to apply for her citizenship? MN1 form, online or paper? Use the council service or not? Ultimately I like her to have her British passport asap.

Thanks in advance for your advice.

1. Yes

2. There is no "short" route. Online and paper can be 6 to 8 weeks or it can be many months. How you choose to apply is your personal choice but many use NDRS with the joint scheme for online MN1 and paper passport form.