What you need to do is;
1. Start the process of making an appeal so that you do not get caught by the deadline.
2. Get NATS HR to confirm in writing that the reason for the height restriction is the perceived problems with radar consoles at LACC, LTCC, MACC and ScOACC.
Having obtained that info;
1. Ask them when the next course to be sent to tower locations such as Luton etc will start and what their reason is for not putting you on that course.
2. Ask them for full disclosure of the study into physical geometry including measuraments that they are using to set the limits.
If the NATS case rests purely on "maximum height" then they have made a mistake. What actually matters is physical geometry. The fact that you are tall probably means that you have longer arms than most. Therefore you can sit further back from the display than the average person but still reach the strip display. With a sloping display, this act of being further away can place you within the normal viewing angle. Your Class 1 medical confirms that you will not have a problem reading text from that distance and modern radar displays have the ability to increase the size of the text in many stages.
Being tall will mean that you will sit with the chair at the lowest settign and probably with your knees slightly higher that your buttocks. This is a far better way of sitting than the one used by short people who have the seat at maximum height and their feet off or only slightly touching the floor.
Overall, when it comes to physical geometry, being tall is far less of a problem than being short because a short person who has arm length in proportion may have problems reaching the display from a sitting position and will either have to adopt an unnatural seating position or (better) possibly stand. They will often be viewing the display from a lower position then expected during HF design and may have to cope with glare from overhead lighting that is not a problem for people of average or higher heights.
Many times I have visited towers and found that the ATCOs (and ATSAs) spent much of their time straining to see over consoles and round pillars because 90% of their time was looking out the window. Being tall in a control tower can have many advantages so perhaps NATS should be persued down that road.
Finally, NATs is not the only training establishment in the UK offering ATC courses........try and obtain an agreement to train you from one of the other providers and then you can;
a) Take it to your bank to get the money; and
b) Use it to bolster your case against NATs.
Best of luck
DFC