Goodness, Dueling Fanjo, I am so sorry that this uncertainty is dragging on for you.
I'd suggest ringing the Midwives clinic first thing tomorrow. Get to speak to one of the duty midwives and let them know that you have had an inconclusive nuchal result and you'd like to be referred for an amnio at the Fetal Medicine Unit. Let them know you've already seen Beattie too, that's the best way to get onto his list (if that would be good for you). Don't worry about crossing over from private to NHS. In my experience, it didn't matter at all. Don't know how far along you are, but if you have decided to go for invasive testing, they do the CVS at the FMU too.
I totally understand your need to know, to have an outcome. I have had invasive testing 3 times, with good and unfavourable outcomes. Each time, regardless of the result, it was a few days of hellish stress, followed by the relief of knowing. Hopefully, then you will be able to relax into the pregnancy, knowing that all is well.
I'm sorry that the Spire clinic didn't quite live up to my hype but I do hope that from now on you find things easy to organise, clear to understand and most of all, that you get the best of news, as quickly as possible.