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The common cause for this type of problem is usually a flow issue due to either you have trapped air in the system or the water pump is weak and not pushing the coolant as it should. Has the cooling system been properly bled?
The best way to bleed this system is with a vacuum bleeder. It work very well to get all the trapped the trapped air out and if it is trapped air it usually ends up in the heater core. It take a special bleeder tool to do this. See it in the link below.http://www.ebay.com/itm/like/252683902211?lpid=82&chn=ps&ul_noapp=trueSee this link for the procedure to do it as well.https://www.youtube.com/watch?v=_WCRcuCZI50
Since the pump is new then I would try having the system vacuum bleed.
If the pump is new and the heater core is not restricted then there really in not much else that can cause this type of flow issue. If it produces heat when you get the engine up off of idle then it has to be a flow issue.
Give this a try and let me know the results and we can work from there is need be.
Are you sure the heater core is flowing freely with no restrictions? Did they back flush it?
It should not have needed back flushed if its new, when the heater stops blowing hot air if you place your hands on both heater hoses at the firewall where they connect to the heater core are both hoses hot or is only one getting hot?
Okay this is something you can check your self, warm the engine up , turn the heater on and when it start blowing cold air go out and check the hoses. If you have good flow through the core both hoses should be to hot to hold. Check this for me and let me know.
Did you ever check both hoses with a warm engine and the heater running to see if they are getting hot?
Okay so you back to a flow issue, If the system has no air in it as you have indicted then you have to wonder why the new pump is not pushing the coolant as it should.
The amount entering the heater core should be the same amount coming out. There should be no restrictions in the core. The problem seems to be with the amount flow you have from the pump. Ford dose not list any flow amounts . With having spent $ 2000 they should have been able to get you great heat. Something is being overlooked here and I am not sure what from here. I will open the post back up for you and perhaps someone else will have something new to add.