Hello Isla and welcome to the Forum
It does seem to have become more complicated nowadays to try and track down a reputable breeder and there is no 100% guarantee that any breeder listed as licensed; a member of the Assured Breeder Scheme or on the breed club lists will be the perfect contact for a puppy for your family or that they are doing all you would expect from an ethical breeder.
I will try to answer some of your questions and explain what is most important in your search. The Assured Breeder Scheme members are required to provide proof that the parent dogs of a litter they are selling have a clear eye test result within the past 12 months although this has of course been impacted by the lockdown restrictions recently and therefore may not have been possible. The litter of puppies should also have been eye screened by a veterinary ophthalmologist to ensure they have no sign of congenital hereditary cataracts but, again the restrictions of the past 18 months or so have affected this too. The eye screening is a test that should be carried out annually on breeding dogs and is an important one, so even if there is a genuine reason for lack of a recent eye screening, you would still want to see evidence that the parent dogs, and further back in the pedigree had more than just a juvenile eye screen. The health test results for both the eye screening and for MAC DNA screening are available online through the Kennel Club health test results finder and, if you have the registered name of the parent dogs, you can check this quite easily. Numbers of litters bred from particular dogs and the inbreeding can also be checked with figures given for the average in the breed to give an idea of what you are looking for. I would always go further back into a pedigree to check the information on the dogs at least two if not more generations back for this information too.
The Assured Breeder Scheme is the only Scheme that currently has experienced assessors who carry out inspections on members a minimum of once every 3 years (again with a proviso re the past couple of years!) and the mandatory health testing requirements. Breeders on breed club lists are not checked by the club and neither are the breeders on the Champdogs website although the latter does have certain requirements which anyone listing litters of puppies do have to abide by including eye screening with regard to miniature schnauzers.
Prices will vary somewhat depending on locality and always have done with the South of England usually higher, however this is not universally so. I would not expect you to be paying more than £2,000 for a quality puppy, from health-checked parents and raised to the highest standards. The greatest likelihood of a healthy, well raised puppy will be from good lines going back through the pedigree and will be from breeders who show their dogs, breed occasionally and have good health and temperament as a priority. You are not going to get a quality puppy at a low price but quality does not mean paying a ridiculous sum of money. Puppies from other pedigree breeds that are numerically rare in the U.K. and raised to high standards are sold for much less than £2,000 from breeders who prioritise the health and temperament and are very careful as to who is allowed to have one of their puppies.
I would suggest using the Assured Breeder lists (Champdogs etc.) as you have been, to contact as many breeders in your travel range as you can by sending a fairly comprehensive e-mail as regards your family, environment, why you want a miniature schnauzer etc. and asking whether you can be considered for a puppy from their next litter. If possible try to find out as much as you can about the breeder beforehand - do they have a website or Facebook page for example? - and do ask about this when you contact them if you have not found it before. You may be able to assess from these whether you feel comfortable with what you see - are they constantly selling puppies; is there information about their dogs as well as puppy sales. It is unlikely that the best breeders are needing to advertise their litters so you will need to get on a waiting list and you do want to feel confident and happy with the breeder as someone you would want to have a long term connection with as regards your future puppy for advice and support - not that they are just selling the puppy and that it is it. Some breeders may be doing everything you would want in terms of health testing etc. but you just don’t feel comfortable with them and the long term connection is an important one IMO. We have had puppies in the past twenty years or so from five different breeders and have remained friends with every one and, with all those who have had puppies from us we have also maintained contact and made some firm friends.
The lack of puppies at present from the best breeders is still due in part to the uncertain times we are living through. Many of the best breeders will not be using a sire they own which means travelling with their bitch for a mating when she is in season - some people prefer the dogs to have two matings 48 hours apart which could necessitate staying overnight somewhere or travelling a long distance four times. During the lockdown and travel restrictions breeders were unable to do this and with a usual 6 month gap between seasons, planned matings may not have happened and, as summer births are preferred to winter litters (due to the difficulties of house training and socialising puppies in the coldest and darkest months) it is not surprising that there are fewer puppies available just now from good breeders.
There are some good breeders in the South / South East area but if you are really committed to getting a puppy from a good breeder, you should consider travelling. We live in Yorkshire and have travelled in the past to the South West and to up near Glasgow to get a puppy from a trusted breeder and have had families who have had puppies from us who have come from NW Scotland, London, Kent and the North West of England - in one case entailing a 5 hour train journey each way for a visit and again to collect the puppy. It may not be necessary but you don’t want to put any unnecessary blocks in your path to get a puppy that has been raised to the highest standards. ( I would just mention that we don’t breed miniature schnauzers and have not done so for several years - our 3 minis are all spayed.)
I would also recommend, if it is possible in the coming months, to try and attend a breed club show or a general breed Championship dog show (Crufts should be taking place in March 2022) where you will have the opportunity to meet breeders in person at one place and see their dogs. Both at Crufts and at other venues Discover Dogs also gives puppy seekers a chance to see the dogs and talk with the people on the miniature schnauzer stand regarding your search and maybe get a recommendation or chance to speak to a breeder there. Personal contact, feeling happy and confident in the breeder is really important and convincing them that you will provide the best possible home for one of their puppies too
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) and I know we will likely have to wait a while longer but I would at least like to work out which breeders might be worth trying to contact as I've read we might have to go on a waiting list. 
