In my opinion, you need warm water for diaper changing. Whether you want to carry a bowl of warm water into the nursery x times a day is up to you.
And in the bathroom, you need a platform where you can dry and dress your child in a back-friendly position after bathing/showering.
In the apartment, we had the washing machine in the bathroom, and a cover for it, which then served as the changing table. (No, spinning is not a problem)
Here in the house, we have a Malm from Ikea; there are attachments for on top (Ebay Kleinanzeigen). Practical and cheap.
By the way, the diaper pail from Chico is worth an investment...
Since we don’t believe in those (very) expensive sets from the baby store (cheapest pressed wood, but nicely color-coordinated, brown, cream, and taupe, a mocking 1200€ for the set), we bought a used one on Ebay Kleinanzeigen. Wardrobe, shelf, and dresser, solid wood, 50€ with self-pickup. Pink. 14 years old. The wardrobe and dresser are big enough to fit larger and enough clothes. Kids’ room instead of baby room.
The crib is the one from Amazon for around 100€ with slatted sides, mattress from Aldi Nord.
We borrowed a bed side crib for the short time.
Briefly about the behavior of (my) baby. For the first days/weeks, she slept on us. Then in the bedside crib. Then in the baby crib in the bedroom. And after about a year, a bit more, in her own room. Except on weekend mornings from six, she comes to cuddle, or when she’s sick. Then also with us.
Your baby/toddler will not stay in their room. Neither for sleeping nor playing. Mother-in-law will tell you that the child’s well-being is at risk if the child does not have their own room, but that is nonsense. So you can also take care of the room calmly in about half a year to three-quarters of a year.
By the way, Lidl has great organic cotton changing bodysuits in solid colors. But that is well-known; you have to be early there.