My experience (as a manager in the HR consulting practice of a leading 'big brand') is that there is good demand for HR consulting skills.
FYI - HR consulting work usually breaks down into the 3 distinct areas: (a) transforming the HR function (b) employee performance (sales, contact centre, IT, finance) or (3) general change management. It may be useful to show you how you where your partic experiences would best fit with this in mind.
And completely second previous post: key will be showing you are thinking about how your insights might translate into actionable, measurable activities with a client. Whilst you dont have direct client experience, it will stand you in good stead if you thought through how you IMAGINE it might work.
All the best,