It sure sounds like the carburetor cannot keep up with the need when the engine is cold and since your carburetor is a primer, not a choke, you have no other way to add extra fuel in addition to pumping the primer. I am going to post a few points that I'm sure you have already checked but here goes. I have had people bring equipment to me that will not start and I drain the old fuel, replace the carburetor and any other needed items, add fresh fuel and the machine runs great. I return the equipment to the owner and they use it until it needs to be refueled. They refuel it and they bring the equipment back saying all kinds of ugly words about me and their equipment as to why it will not start only to find out that the fuel in their fuel container is from last year and has gone bad. I then ask them how that crow tastes. Actually, I only think of saying that and I educate them on the need for fresh fuel and I get to charge them again for the additional needed work. So, I'm going to ask you the same question, are you absolutely sure the fuel is fresh? By fresh, I mean 30 days or less. If so, then are you absolutely sure the replacement carburetor is an exact match of the previous one? If so, is it possible the factory messed up and put the wrong jet in the replacement carburetor? Did you remove the bowl (if equipped) from the carburetor and have a look for any contamination?
Were the carburetor mounting surfaces absolutely clean and did you use new gaskets? It is possible an air leak exists causing a lean condition when the engine is cold?
Please tell me what the "other" issues were for replacing the original carburetor.
Do you use any fuel additives such as a fuel stabilizer?
It sounds like there is a fuel restriction somewhere preventing the proper air/fuel ratio for cold start operation. Have you tried pumping the primer six times and starting the engine cold? How about eight times? How about more than eight times? What were the results?
If you will answer all the questions I have asked, we will narrow down the possibilities as to what is causing the problem you are experiencing with your mower.
At this point the cause of the problem could be a lot of things and even something I haven't mentioned yet so I am starting with the basics and doing the best I can with the limited information you provided.
Take your time in checking things out and report back with your findings when you are able.