Paper
Well this was a lot harder than I expected...
I'm going to focus on plain paper here, partly because lined, dot or grid seems very hard to find, but also because that's what we used for over 10 years - boxes of plain paper used to print diaries, templates and lined/dot/grid pages. Diaries and templates were always printed in-house on a laser printer, and most A6 coloured dot/grid and line pages were also printed this way - it was only when volumes of A5 and some A6 were much higher that I got the pages printed externally.
I have made the grey lined, dot and grid layouts available for download and printing (see the Layouts/Templates page), but it may also be easier to just print one of those as a guide sheet underneath plain paper.
Paper from other discbound systems (Atoma, Levenger, etc) should be compatible with our notebooks, but you may need to trim the size a little - for example the Atoma paper is A5+, which means it fits on the rings perfectly, and is the right height, but it is a little wider than A5 and so sticks out of the side when the notebook is closed. Trimming the paper down to 148mm using a guillotine would fix this issue.
For non-fountain pen users, there are options - a quick trip to Amazon, or a few minutes spent Googling will yield A5 plain or even lined paper that can be punched to use in our notebooks. You may even find limited quantities of A5 discbound-punched paper from time-to-time. I have not found A6 Lined, Dot or Grid paper available here in the UK, but A5, or even A4 plain paper, cut down to A6 size using a guillotine is easily done.
For fountain pen users, it seems a little more tricky. There are a few brands of well-loved paper available - Tomoe River, etc - but sheets can be quite pricey. I'm not going to list any of those here because I have never used them myself and their availability frequently varies, but a question or search in fountain pen communities will find the current recommended options.
I have found and tested a couple of options here in the UK, and they may be available in Europe or the US too :-
Clairefontaine Triomphe Writing Pad (50 A4 Plain Sheets ) 90g/m²
This is a really lovely paper, and works great with a fountain pen. It comes in pads, and you have to remove the sheets from the pad, but that's easy and leaves no residue. I used the A4 version, cut down to A5 and punched, and it's lovely. At the time of writing, that pack cost me £5.53 from Amazon, so 50 x A4 sheets, cut to 100 x A5 sheets would be a bargain - even more so if you cut it down to 200 x A6. I have always heard great things about Clairfontaine paper (and its sister-brand Rhodia), and it's probably the one I would use myself.
HP Premium Printer Paper (250 A4 Sheets) 100g/m²
Right back at the beginning of William Hannah I read a comment from a user that they always used HP Premium paper for writing with fountain pens and it was fine. So I just ordered a pack of A4 to test, and it's pretty good - not perfect, but good. There is very minor feathering, and very minor bleed-through, but as a cheap day-to-day paper I would happily use it. Not sure how it would cope with wider nibs or wet inks, but with a Fine nib or Gel pens it works well. And at £9.59 for 250 A4 sheets it's incredible value.
I'm sorry I haven't been able to find many specific options to replace our paper, but with a guillotine and a punch it is relatively easy to take any good paper you find, and use it in your notebook. I encourage everyone to join the William Hannah User Group on Facebook - there are already a couple of posts about good, alternative papers, and the more the community shares, the easier it will be for everyone.