It'll probably take about two weeks to get a referral to urology, and then perhaps six weeks for the tests, and another wait for the results, so you're probably looking at 2-3 months all in. It's the hardest part of the whole experience, unfortunately.
Generally you'll first have an MRI scan. If the scan shows areas of suspicion you'll then have a biopsy. If the biopsy finds cancer cells you'll have a bone scan to see if it's spread to the bones (which, with your PSA it almost certainly hasn't, but it is a standard test). Each test usually takes a couple of weeks to report on, so unfortunately it all takes time.
Once all the test results are in, all the information will be discussed by experts in all the relevant fields at a weekly multidisciplinary team (MDT) meeting, and it's only at that point that treatment recommendations will be made.
About half of men in their 50s do have prostate cancer, by the way. I was diagnosed at the age of 55 in 2018. Three years on and I'm all treated and good to go.
Best wishes,
Chris