I lived in England for about a year - a month was what I also expected (at least) and also what some of my friends who had the experience told me. However, I called them on a Monday, got an appointment for Wednesday, and my NI number arrived in the mail by next Monday - so it actually took me a week.
I'm sure it depends on the time of year and other factors as well - I moved in December, so maybe it was a quieter time? Also, take into account that the offices in London might have longer waiting times because it's a large center - I had my interview in Cambridge, which probably made it easier.
The process in itself was quite simple - I'm from a EU country as well - I rolled up in the office with my ID, they asked me to fill in a form (usual basic information, questions about work, where I'll be living in the UK, plans, some safety stuff), they made a copy of my Estonian ID, and off I went. If you're a regular person with no horrendous criminal past or ill intentions while staying in the UK, they're pretty quick with processing and sending out the number to you. I'd say the entire process including waiting in line took me under two hours?
Oh and also - just FYI - I always thought you'd get a NI number card, but they don't actually give out physical cards anymore - you'll just get the number. I tried for a few hours to contact the NI number call line just to finally receive info that physical NI number cards are not a thing anymore