In my country (Ireland) the funeral generally takes place about three days after the death. It may be delayed slightly if there's a close family member living abroad, to give them time to get home. To us it seems really odd to have a family member die and then have a to wait a month or so for the funeral and burial. And I'm sure our way seems equally odd to those of you used to waiting several weeks.
How does it work? You immediately start ringing around family and friends, to let them know. You contact the priest immediately and you also contact the funeral home. When my Mam died, the lady from the funeral home was amazing. She gently reminded us about things we might have overlooked. Between them, you work out the funeral arrangements. The funeral home take the details and make arrangements to put a death notice in the newspaper and also on rip.ie (website for death notifications). My parents had already purchased a double grave, so we just had to phone the graveyard to notify them that we would need to have the grave dug on x date.
The family and friends you have notified will check rip.ie for the funeral arrangements, so you don't have to contact them again to let them know the arrangements. Everyone knows that the funeral will take place quite quickly after the death. Funerals are open to all here - it's not like the UK where it seems to be invitation only (or at least, that seems to be the case in England). In some rural areas, the death notices are read out on the radio.
You choose a pub for the obligatory soup and sandwiches for the mourners for after the funeral.
You choose family members to do readings or prayers, contact them to ask if they're willing to do the reading/prayer and then arrange to have copies of the prayers available for them on the day, to take up on the altar.
You choose gifts to bring up to the altar. These are usually related to the deceased members' interests and hobbies e.g. a jersey (if the person played a sport or followed a team), an instrument (if they played music), a handbag (if they were into fashion), a family photo (to represent their love for their children or grandchildren etc.). You choose family members or friends to carry the gifts up to the altar. When the gifts are being brought up to the altar, the priest will say a sentence or two to explain their significance.
One family member is chosen to do the eulogy at the end of the mass (assuming the priest allows it - some priests don't) and will usually spend a bit of time writing the eulogy or writing notes to help them.
Some families have a wake. The body is on display in the coffin in the house and friends/families call in, to say their last farewells to the deceased before the coffin is sealed. The family have to shop for tea, coffee, sandwich ingredients, cake etc. and prepare sandwiches for an army of mourners who are likely to call to the house. The extended family and neighbours often help with this - neighbours will send up trays or tins of sandwiches. Friends and neighbours will also help to wash up (the cups and plates will be in constant circulation - no time to put the dishwasher on) and may be called upon to nip to the shops if milk or sandwich ingredients are running low. There may be set hours for callers to the house - this will be specified in the death notice. If the family don't want any callers to the house, this will also be stated in the death notice.
The period between the death and the funeral is an intense period of activity. If you're working, your employer will understand that you won't be back at work until the funeral is over. It's not uncommon to take a week or two off if a close family member has died - a few days Compassionate Leave and a few days of Annual Leave tagged on. You won't have time to really register the loss or start to grieve until the funeral is over.