I get the impression that sleep Apnea is pretty easy to diagnose, but getting it done through the NHS is ridiculously difficult.
I think, like many things with the NHS, it is dependent on your GP and to an extent, the consultant/specialist you see. I've had multiple sleep studies through the NHS, and getting them wasn't particularly difficult. The main thing was the waiting that's typical of non-urgent referrals to specialised services. (though if you're significantly sleepy, you can probably get an urgent referral if your sleepiness could cause danger to yourself and/or others - usually for people who are drivers by profession or operate complex machinery on a daily basis)
Probably the biggest hurdle will be convincing your GP (depending on your relationship with her/him) that your sleepiness is something beyond the usual suspects (like stress or anxiety) and that it's to a point where it's disruptive to your daily life. Before a referral to a sleep specialist, you might have other things checked first like iron, thyroid, and B12 to rule out organic causes of sleepiness (disease, vitamin deficiencies, etc.). You might be asked about lifestyle, sleep hygiene, and mental health. Improving sleep hygiene is something you can do yourself, everyone should generally strive for good sleep hygiene anyway, and if you do it before seeing a GP, you can say that you've already covered that when it likely gets mentioned.
Also, I highly recommend doing a sleep diary for a week (or more if you can) that details what time you go to bed, the time you think you actually fell asleep, the time you wake up, and how many times (if any) at night that you wake up and for how long. In addition, you can self score your level of sleepiness with the Epworth Sleepiness Scale to give yourself and your GP an indication of how severe your sleepiness is.
