Sorry for not getting back sooner but have been on the allotment all morning.
To tell you the truth if you don't know the manufacturers name and the model number of the tap, it is always much quicker to replace the taps if they are quarter turn (cassette).
The down side is .. if you do manage to separate the head from the body and you take the cartridge along to a hardware store, they may have one or two odd cartridges in stock. The chances of them having the right amount of splines and finding the right length spindle, not to mention the diameter of the spindle. If any of these measurements is wrong, the new cartridge is unlikely to fit your existing tap head.. If you take it home and try to fit it, mark or damage the cartridge and the store 'want it back.
The best plumber I know told me twice and I have stuck by that rule. .. "If you get a quarter turn tap that you cant identify, never try to fix it.. always replace it."
(The first time he told me I ignored him and tried to replace the cartridge a few hours later I still hadn't found one)
There are always BNIB quality taps on eBay for a fraction of the cost.
Good luck !