I went through the graduate recruitment with Arup back in 2017 and was offered the Graduate Electrical Engineer (Buildings) job to work at their London office.
I applied early November, did all the online stuff as you mentioned and received the first communication back late November (so about 3 weeks later).
Another week later a careers coordinator got in touch with me asking to confirm my attendance for the interview (assessment centre) held in mid-December.
Things might have changes since then, but this will give you a rough idea on what to expect. Don't be afraid to send them an email and ask for an update, if anything it shows that you're genuinely interested in the job.
I did get the job offer in the end, so ask me anything.