I replaced the ink cartridges on my Epson C40UX and after several cleans and nozzle checks only the blue and yellow inks are coming out
I presume your printer was printing all colours before the cartridge change. If you removed the old cartridges a few hours or more before you fitted the new ones, it is very likely you have nozzles clogged with dried ink. If this is the case, do a head soak as follows. Turn the printer on, and as soon as the print head moves out of its parking position, pull the power cord out of the back, or turn the printer off at the mains. You may now carefully move the print head by hand over to the left hand side out of the way. Look inside the print head parking area on the right and you will see a small rubber cup that comes up to cover the print nozzles when the head is parked (to stop it drying out). Use a syringe or dropper to fill this cup with head cleaning fluid or distilled water. DO NOT USE TAP WATER. Carefully push the print head by hand fully into its parking position so that the cup full of fluid comes up over the print head. Do not turn the printer on at any stage in this process. Leave this for at least 24 hours, then turn the printer on and try a head clean/nozzle check again. If you now get any of the missing colours on the nozzle check, then there is hope! Try 2 or 3 more head cleans (but no more). If still not a full print, repeat the soak process until you get a full nozzle check.
If the new cartridges went in immediately after the old ones came out, it is unlikely to be blocked nozzles. If you changed the cartridges without using the proper procedure with an ink charging cycle, you may well have got air into the nozzles. This just takes patience - try a daily head clean to see if the ink gradually appears.
Finally, you may have faulty or out-of-date (dried-out) cartridges. The easiest way to check this is to try new cartridges in the nozzles that don't print, making sure you use the proper cartridge exchange procedure. Compatible cartridges are very cheap on ebay - £3 a set in the UK.
Please come back with further information on what you have tried if none of this helps.