I've had NHS aids for 20 years, and been prodding my MIL to use aids for 12...
The only reasons to go private are to get the tinier more 'discreet' aids, and less waiting time. However, no-one ever notices hearing aids anyway (seriously. mine have magenta earmoulds, shiny metal rings on the tubing, and black aids behind the ear, and people still don't notice!), so I'd advise against tiny in-the-canal aids because they're bloody fiddly.
Your mum needs a referal to Audiology for a proper hearing test - either via GP or try phoning your local hospital's audiology unit and seeing what they say (they may also say that hospital X down the road has shorter waiting times!). Your mum probably has standard age-related hearing loss, which means they have lots of suitable aids on the NHS, and they're pretty easy to program in one appointment.
So I'd start with the NHS - everything coppertop says has been true for me, although I've got a complex loss (head audiologist looks at tearful student plotting my hearing and goes 'what the f* is that?', looks up and sees me in the testing booth and says 'Oh, it's you again'! - cue three audios spending 3 hours tweaking my aids!)
However, aids aren't a magical solution, especially if your mum has spent 30 years not being used to hearing lots of things. You need to retrain your ears to ignore certain noises and get used to understanding other ones. And to changing program when there's lots of background noise, so they only pick up what's in front of you. My MIL will now wear one aid to use the phone and watch TV, but the rest of the time it's a case of training her to ask for repeats and FIL to get her attention before speaking and state the subject of the conversation up front.