The first thing I'd point out here is that printers without scanners are
(a) quite rare ; and
(b) often a great deal more expensive than printers with scanners. (I assume that's got something to do with the volumes of production). So please don't rule out a printer with a scanner. You might find that it's cheaper to buy 'with', rather than 'without' anyway.
It can often work out cheaper in the long run to buy multi-cartridge inkjet printers (which use 5, or even 6, cartridges) than to buy printers which just have two cartridges in them, as replacement inks can work out far cheaper. A pair of replacement cartridges for a cheap two-cartridge printer can often cost over £40, even using 'compatibles', whereas a set of five cartridges for a multi-cartridge printer can cost under £15.
I can't find any inkjet printers that leap out at me as being ideal for you at the moment but I'll keep looking.
However, if you only ever need to print in black and white, a mono laser printer might be ideal for your needs, with this being the obvious first choice candidate:
https://www.argos.co.uk/product/1946417
Compatible toner cartridges for it cost just £15 which, at 1000 sheets per cartridge, works out at just 1.5p per sheet:
https://www.stinkyinkshop.co.uk/brother/toner-cartridges/brother-hl1112