As SoonToBe60 said, bit hard to say without more information - have you injured yourself because of it; are you allowed to drive (GP would have given informal recommendation even if you’re a non-driver when they did the referral); do you feel safe having sole supervision of children (especially at bath time, for example)? Are you making, or at risk of making [serious] errors at work; & what would the potential consequences be? (Both the eg killing someone/losing millions/having to re-enter a load of data/awkward call to client side and the warning/full disciplinary/fired side.)
Exhaustion does cause functional impairment & there’s a reason sleep deprivation is used as torture. There is a risk, though, that the time off wouldn’t help as much as you hope, depending on the exact issue. It’s also the case that given what’s happened over the past year everyone is feeling worn down & exhausted - & while you absolutely might be feeling 10 times worse as you’ve an underlying health issue, it may be worth considering how taking the time off would be received in your workplace.
The other thing (& I really really do not want to sound all Voice Of Doom here, but think it’d be wrong not to mention it) is that you might want to hold off on this option until you absolutely hit The Wall. I mean, hopefully you won’t get to being unable to think in complete sentences or focus well enough to complete fairly simple tasks without prompting or get so exhausted by basically heating food you’re too tired to then eat it - but obviously then, or slipping into that, you’d absolutely need time off. If you’re still in a space where things are copeable with, maybe looking up resources on conserving your personal energy (eg shower at night sitting down so you don’t start the day doing something that eats energy) & pacing to try to help you manage the exhaustion. If you’ve not yet been sent an appointment date, it’s perfectly reasonable to contact the hospital with a polite enquiry as to when you might be seen. They might not be able to tell you, but they’ll usually try to give some idea of what the wait is looking like for you.
Ok, after writing my v long post I checked back to see if you’d replied. If it’s sleep apnoea they suspect, a couple of weeks off probably won’t cut it - as you say, you’ll very quickly land right back where you started. (There’s something called central sleep apnoea, where your brain basically forgets to breathe, rather than it being weight-related as obstructive sleep apnoea usually is - & it doesn’t present in exactly the same way. It’s good your GP has been proactive rather than dismissive though.) It doesn’t sound as if your work would be super-supportive, either - & a bit as if you’d potentially be putting yourself in a precarious position employment-wise, even if thinking more long-term (sorry if I’ve completely misread/misunderstood that). Trying out any of the various lifestyle changes suggested by reputable sources that are applicable & accessible to you might be worth it? Even if they only help a wee bit, a slight improvement is still an improvement. Making sure - as far as possible - you’ve not any other causes of fatigue adding to the problem is the other thing. Not suggesting sticking your children in a cupboard or something, I mean taking a Vitamin D supplement because the whole of the UK is a bit short on that; B12 if you’re vegan/vegetarian; & tbh a decent multivitamin won’t hurt you but will make sure if you’re short on anything & it contributes even slightly to your feeling grim, you’ll fix that bit.