I've had private clients before, however I don't take them now even if approached. Some thoughts:
The pool of employees who are in a position to pay for this advice is going to be small. Employees are more likely to wait until they have no option but to pay for assistance and by then most of them would opt for a solicitor.
Many people have cover through their house insurance for employment disputes, which would provide them with access to a solicitor when required (often poor quality and with no choice of provider, but still). This cover wouldn't extend to an HR consultant.
I can't imagine there are many employers who would allow staff to bring an external HR consultant to a hearing.
There is very unlikely to be repeat or long term business from clients, or any opportunity to upsell or sell different/additional services which makes marketing inefficient.
They are unlikely to be in a position to refer you business, however much they might like to, which again makes marketing inefficient.
Private clients are generally harder going than corporate clients, they are more intense, they are (understandably) generally stressed, taking things very personally (again perfectly understandably), and this all comes across while dealing with them.
You won't be able to charge as much (or won't feel you want to charge as much) as I expect you do for corporate clients.
My private clients all came through seeing me on MN, so there was no marketing involved, but I've been in a position for quite a few years now that I can pick and choose clients, and the first thing I did when that happened was say "no more private clients".
They were all very lovely and very pleased with what I did for them, we got some really good results and I have some wonderful testimonials from them. But from a business point of view my time is much better spent on corporate clients.
That is not to say there isn't any market for those services, and I am throwing all the negative at you purely because I think that's probably a useful thing to do, so that if you do choose to go ahead, you've got as clear a picture as possible of the negatives so that you can address them early on.