Ask an Immigration Solicitor. Get an Answer ASAP.
Good evening and welcome to JA.
It is correct that most employers now requested a BRP card to prove your right to stay in the UK.
You will need to make an application to get an ILR in BRP format.
You can also apply to be naturalised as British if eligible straight away without first obtaining BRP. please let me know if you have further questions and I am happy to help.
Regarding your family members, it is always best to get ILR in BRP format as stickers in the passports are not used anymore.
Once you apply for a BRP you may request a premium service by which you will get a decision within 24 hours but you will need to pay extra fees for that.