I am unable to obtain any water flow from the cold water side of a new single lever Mixer tap

hot water pressure is equal to cold water inlet pressure
check for tap restriction, hot water section free of air , ( release the pressure valve ;lever to fill and bleed out air)
if you have a hot/cold mixer at the hot water system, they can fail and block of the hot water flow
(That unit has to be fitted by law on heater installation)
have it tested first as they cost in excess of $15.00 aud to replace

check other hot water taps to check hot water is flowing from the hot water system(tap on heater not turned off)
check faucet handle is put on in the right place --only 1/2 movement to the stop
check for blockage to the hot water side of the insert ( incorrectly installed )
still with problems --call a plumber

problem in the tap washer (expanded and blocking the water flow
blockage inn the connection where the shower head connects at the wall
hot water flow restrictor so that you will not get burnt by the hot water --test by turning on the cold water first and then opening the hot tap as well

Sounds like water leak.
For example, broken pipe... or water running out of TP valve.
That would explain variable hot-and-cold water at tap.
Turn off all taps, and observe if water meter is registering water usage.

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you make no sense. From the picture the water "mixer" is the center lever or is it a diverter valve for the shower and water nozzle below???
The knobs are for hot and cold water??

Start with the pipe that extends out from the wall where you hooked up your cold water connection.
Get your self a large container and many rags or towels.
Gently loosen the connection to see if there is any water coming out of the wall pipe.
If not, there is your problem.
If so, you have something plugged up or wrong with the hose or the faucet.
Remember, normally a person will turn off the water at both valves under the sink when the replace a faucet.
Those valves are where most of your connection hoses go.
A valve can be broken or stripped, so that it will not turn on after you have turned it off.
God bless your efforts.

In a word no. Combi boilers work on balanced pressure.ie the mains cold supply to the boiler should be around 1 bar minimum which would give a hot water pressure of the same 1 bar. Assuming the cold water pressure is fine, it sounds like you have a restricted flow on the hot side. If you live in an area with water problems the hot water systems pipework calcify over time effectively reducing water flow. Unless any flexi hoses are kinked, check if there are any filters in the appliances that may be blocked.Also check all valves are fully open and if possible check if there are any restrictions at the valves. Failing all that and the taps are in good order it is possible to fit a pump before the boiler but this may only be a temporary fix overall.

you may just have to replace the hot/cold stem

Remove the tap outer cover. Take out the barrel & inside there is the washer on the end of a screw which slides in and out when the tap is turned. This could have been turned too far out and will not go back in. Clean both the screw thread and inside the barrel, lubricate with petroleum jelly or similar and reassemble & test before replacing the barrel. Again lubricate all threads.
Alternatively buy a new barrel

hi you 1st need to isolate both hot & cold water supplies to bath.lift the lever up & underneath the lever you will see s screw unscrew it & be careful not let it go down the plug hole, so a good idea at this point is to put in the plug!!!

Once the screw is loosened you will be able to remove the lever.
then there is a cover which will need to be undone & removed seeing as its gold you will have to be careful undoing this as it may be stiff, sometimes a few taps with the endle of a srewdriver handle will loosen it enough so you can remove it. once this is removed you will gain access to the hexagonal part of the cartridge, so yo uwill need to have the appropriate spanner or adjustable spanner. you will need to hold against the tap when undoing it so as the whole tap body doesnt move.once the cartridge is removed the new unit can be fitted in reverse order.it sounds simple but being Gold you have to be careful you dont damage it whilst attempting to remove the old one. Tip put some silicone gtrease around the o rings on the new unit as it will make it easier to remove if you ever have to remove it again?

Kindly

Richie