It depends, I'm pretty sure that if baby Lorena's health problem and cause of death was known, they won't do an autopsy and the death certificate will take about a week or two. If the cause is being investigated, this will take a bit longer.
You can usually plan the funeral (service, coffin, flowers etc) without a death certificate, but not set the actual dates. The body can't go to the funeral home until they've got the death certificate.
It depends on if you know what you want in terms of the service ahead of time. They will talk you through coffin choices, music, songs, hymns (if applicable), orders of service, invites, cremation or burial and in which cemeteries or crematoriums, headstones, and then if it's a church service the vicar will talk to you about which bible passages etc. Some people know what they want and get through this quickly, other people take a long time debating their choices.
We got our dad's funeral done in less than 3 weeks because we were rushing to avoid the threat of covid restrictions. Also, he couldn't be embalmed which they recommend after 2-2.5 weeks. But most people take a bit longer, and we did feel very rushed and stressed. I would say most people take 4-6 weeks to arrange a funeral.
But then I've known 3 people who've had Pure Cremation services that are mega-quick and fuss free, I've known two elderly men get it over and done with very quickly for their wives with this. I doubt they would do this sort of service for a baby though.