Yeah, Charmed is right, your Avian Vet is not at all one of the best in the UK if they are telling you they have to put your Budgie under any type of sedation/anesthesia to do a simple blood-draw from his neck to run blood-work/tests...That's a very common "red-flag" that a Vet is very inexperienced with birds, or doesn't feel at all comfortable with handling/treating birds, or both...Birds/Parrots have trouble coming out of the sedation/anesthesia, and taking that risk just to do a simple blood-draw from one of the bird's 2 jugular-veins in their neck is ridiculous; I've done it hundreds of times, and I'm not a Vet at all, and all it requires is simply "Toweling" the bird, gently putting a bit of mineral-oil or vaseline on your finger and using it to move the feathers away from their neck area, and then you find one of the Jugulars and you draw the blood, the whole process takes less than a minute...And for a Budgie you cannot draw more blood than 2ccs/mls because they don't have anymore to spare...So I would be very, very leary of that Vet and anything they told you...
The other "red-flag" is that your Vet didn't even run Fecal Testing!!! Yet he's "stumped" about what is causing your poor Budgie's blindness and feather issue? You can't be "stumped" about either of these things until you actually run basic, simple Diagnostics...The first thing needs to be you bringing in a fresh Fecal-Sample (within 24-hours of him dropping it if you put it in the fridge right away and get it to the Vet within that time), so you don't even have to bring him with you for Fecal-Testing!!! They need to run: #1) Culture/Sensitivity, which is usually sent out to whatever laboratory they use, and usually takes 3-4 days to get back, #2) Fecal-Smear on a slide is made and they look at it under their own Microscope in-house, on the spot, and check to make sure he has enough "Beneficial Bacteria" throughout his GI Tract and to check for Yeast/Fungi, Protozoa, and other Parasites, (One of the #1 reasons a pet/captive Parrot loses feathers or starts the process of plucking/self-mutilation is due to a Systemic Fungal/Yeast Infection that spreads to their external/topical Skin and into the Feather-Follicles), and #3) They then Gram-Stain the slide to look for the presence of harmful Bacteria...
So they can run full Fecal-Testing without your Budgie even having to come to the office, and this might be all that is needed to diagnose why he's losing his feathers and not growing back in new ones. It's very common for a Yeast/Fungal Infection to form all over their skin (which means it's everywhere else too), and the first thing that happens is that the bird gets extremely itchy and uncomfortable, and may or may not start plucking...However, if left undiagnosed/untreated, it can actually damage the Feather-Follicles and cause the feathers to not grow back in. So full Fecal-Testing is ALWAYS the first step in diagnosing pretty much any and every Avian health/medical issue..
Then of course he needs to have full Blood-Work done, which again is easy, quick, and NEVER requires any type of sedation or anesthesia, even with the most non-tame bird; It only takes a minute or two from start to finish at the longest, doesn't hurt the bird at all, and this is VERY IMPORTANT because a simple CBC and Liver/Kidney Panel will tell you so much, it will tell you if he has an infection anywhere in his body, if he's Anemic, if he is suffering from any specific nutritional-deficiencies or malnutrition (another very common cause of feather-growth stopping!), any Metabolic issues, and very important his Liver and Kidney function and health...So obviously with both a feather growth issue and a vision issue, this is hugely important and will tell you so much about what it might be and what it is not...
This is very basic Avian Medicine 101, and it's exactly what I do when a bird comes into the Rescue with either Feather-Growth Issues or they are Pluckers/Self-Mutilators, or when they have any Vision issues. So the bottom-line is that your Vet cannot possibly be "stumped" because they haven't run a single diagnostic test yet, not even the most-basic ones that will probably diagnose both issues pretty quickly, or at least tell you what other diagnostics need to be run to confirm the issues...So your Avian Vet is not competant at all as an Avian Vet, and there are a lot of members here from the UK who have decent Avian Vets that they like and trust, so you need to get with them and figure out where the closest CAV or Avian Specialist to you is that knows what they're doing, because your poor Budgie could be suffering with blindness and baldness uncecessarily!
***I'm sorry this Vet did nothing at all to help you or your Budgie, but trust me, you're actually much better-off that this Vet didn't do anything at all as far as diagnostics or anything else, because chances are he would have killed him...