It might be worth trying to find a local writers group, whose members have had some experience of publishing similar material and getting some face to face advice from someone who has done it. For me, living in Winchester, the Hampshire Writers Society would do the trick - not sure about Yorkshire. There are basically four routes to publication:
1. Find an agent, get them to accept your manuscript, they will then find a publisher, the publisher will provide all things needed to bring the book together and publish it (eg editor front cover artist, typesetting etc etc). And the publisher does all the stuff related to distribution and marketing. This is really the only way to get access to the 'big' publishers. The publisher pays the author, and the agent takes a cut of that fee. The author never puts any money into the process. It's hard to get an agent interested in the first place - typically they each get a thousand or more prospective submissions each year to consider (at least for novel fiction and I would imagine children's fiction is similar). Agents tends to specialise, eg romance, children, young adult, science fiction etc.
2. Find a smaller specialist publisher, or an editor at a specialist publisher. The publisher will provide all things needed to bring the book together and publish it (eg editor front cover artist, typesetting etc etc). Again the author does not pay the publisher, and the publisher does the distribution and marketing.
3. Use a vanity publisher. Here the author provides a payment to the publisher that covers the costs of preparing the manuscript, editing, cover art etc etc, so the publisher has much less incentive to push the book in the market place. There is often little or no marketing. The author costs can be hundreds or thousands of pounds.
4. Self publish. You, the author do all the work. You create the book, assembling the contents, edit it, add illustrations, provide cover art, and basically produce a PDF. It is then easy to place this on Amazon. Costs are negligible if you have a computer and some document editing skills. The quality of the book is of course highly dependant on your skill level. Marketing is up to you. Amazon provides print on demand, so you can get copies of your book as needed, within a few days. It will however be very hard to get your book onto the shelves of traditional bookshops, without a lot of effort your part. But you can make sales at book fairs etc. I did
this with some poems I wrote as part of a Creative Writing course, and I found the process relatively easy. Sales have been zero though, not unsurprisingly.
People get good or bad results using any of these routes, there is no one size fits all.
It depends a lot on what you want to achieve. If you would like to simply put your mother's material together into a nice, professionally presented book format, and distribute copies to family members and friends, then it may be simplest to go for the self publish route - where you can simply print a couple of dozen copies on demand.
My suggested first step would be to go a local library and see if they can put you in touch with a local children's author or a writing group and then go and talk to them. Having spent a lifetime in computing, and then turned to creative writing, I have found that writers are surprisingly and actually many times more friendly than computer scientists so you will find a warm welcome I'm sure. (It's the software engineers who tend to stuck typing away on their own at a computer terminal all day - the authors are more likely to be in the pub 'doing research'....)
