Hi, I can't advise on where to go for a 2nd opinion although there are a lot of people who often come on here who have sought out what they believe is the best treatment both privately and in the NHS. My own opinion is the NHS hospitals have big facilities with a lot of back up and in general, if it is private and they allow private treatment, are cheaper than Private Hospitals.
There are places in each region that are classed as major cancer centres. In the North West of England where I live the Christie in Manchester is considered the prime centre. Although there are probably 8 hospitals that do treatment in the region. In Scotland I don't know but you would think Glasgow could have a major centre. I noticed a place on line at BMI Prostate Centre Glasgow although it only mentioned 2 Consultants and they didn't seem to be surgeons and I know nothing about it.
You might also ask what is a 2nd opinion as it could mean another psa test, biopsy and an MRI. If you've just had one set they might be reluctant and it might not happen in a day and take longer for results. Although if it's just a consultation using existing information it might be easier and less time consuming. I assume you have confidence in the doctors you saw and wondered why they suggested a 2nd opinion.
If you reply it keeps your post nearer the top as they slip down which can mean they get less looks.