PPCS on Devonshire Place (London) are good for assessments. You'd probably need to see the psychiatrist a few times to get a full picture and meds sorted if you are bipolar.
The cost when I last used them (2014) was £500 an hour. I think I went 8 times over the course of a year, as well as seeing a therapist weekly. Once I was stable my meds regime was passed to my GP and I see my therapist monthly now just to keep me on track (I pay for her out of my own money as I wanted to chose my own and privately you tend not to be able to chose).
When I have used my work insurance for MH issues I had to be referred by my GP. However you can just refer yourself privately.
Work did not use it against me, they didn't get involved at all. I dealt with Occupational Health who simply provided work with a statement that I was not fit for work and then later advised them on a phased return. They gave me an option on how much to disclose, I was described as having 'depression and anxiety' to work, which was what I was suffering. My bipolar was not mentioned.
I am bipolar I, I do not declare it. That is my own personal choice. If I felt that I needed to declare it I would. If declared I believe you are covered by the disabilities act, which protects you. It really is up to you to decide what information you disclose.
Wishing you all the best.