Most (all?) printers won't print unless all inks are available. The usual reason given is that it can damage the printhead (although I don't actually understand why!).
Some printers
mightallow you to print with only a black cartridge by going into 'Properties', when selecting 'Print', and choosing 'grayscale printing' (with that horrid US spelling!).
If time allows, I can recommend this supplier, who'll sell you a full set of 56 cartridges for £11.22:
http://www.inkredible.co.uk/hp-photosmart-6510-e-all-in-one-ink-cartridges
(If you order this evening, they should be with you on Saturday).
Otherwise, given that places like Staples and PC World are about to close, you'll either have to phone around your local supermarkets to see if they stock the cartridges you require or beg for the use of a friend's or neighbour's printer.