Cannot bleed coolant system

I agree with your logic. Attached vid addresses bleeding. Sounds like at low rpms, not enough flow to transfer the heat. So bubble or maybe a partially obstructed core. There are a couple of vids on clearing a core... GOod luck, let us know if you figure it out.

air in the system need to bleed it,do you know how to do this.

Get into a shop Soon or-- a dealer-- if money is no concern

Snow & Cold time is here for most folks

Need to have the ENTIRE vehicle in the best condition
possible-- for the winter --reguardless of cost

bleed cooling system, probably there is air trapped in the heater core preventing warm coolant to circulate into the heater core. try this
1 open radiator cap fill coolant and put heater in hi heat
2 start engine and wait until you see the coolant start to circulate
3 squeeze hose several times watching if coolant volume decrease
4 add more coolant and repeat 3 until coolant volume stays same or start overflowing.
5 check if heater giving off warm air or repeat 3 and 4 until warm air blows on heater.
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or find the coolant bleeder screw, do 1 and 2
open bleeder screw until coolant come out, do this several times until no air comes out when opening bleeder screw, fill coolant in radiator.

if the cooling system is not clogged and the water pump is ok then there may be air trapped in system, bleed air from system if need be. if air is trapped in system you will not get any heat. start car let it run take cap off of coolant tank then find bleeder for the coolant system, open bleed screw and air may be trapped in system.make sure coolant is at a proper level. do not forget coolant tank cap when done.

The water pump failure sounds fishy to me. I avoid dealerships at all cost. They tend to overcharge , and fix unnecessary stuff.
The timing belt on the other hand should have been done at 100,000 km / 60,000 mi

the thermostat (on bmw replacement parts) have an arrow that has to be installed facing upward.if this was done correctly,to bleed the system usually there is a bleeder screw on or near the expansion tank(where you fill the coolant) and possibly on the thermostat housing. i would recommend opening both bleeders and keep filling the system with coolant closing the lower bleeder first and then allow the rest of the air purge from the other bleeder screw.make sure the coolant stays full and be patient you can't wait TOO long.

It sound's like a problem with the water pump, thermostat, or drive-belt, drive-belt tensioner and pulley, check the tension on the drive-belt, if it has good tension check and or replace the thermostat, if your still having cold air coming out at low rpm's it could be a bad water pump. you can check the water pump by looking to see if there's coolant leaking from the weap hole, or listen for a squealing noise from the water pump when the vehicle is running, there's also a possibility of air in your cooling system which usually occur's when you have the coolant system opened up for repair's, if there is air in the system it could keep the coolant from flowing properly, only way to get it out is to bleed it out using the bleeder valve, im assuming the coolant level is at the proper level,
I hope this was helpful. Good luck.

possible bad heater core

There are known problems with this vehicle's heater core not getting enough coolant flowing through it to blow hot air.
1) Even if the coolant looks good, there are problems with the heater core plugging up. One solution is to use a heavy duty cleaner (CLR pipe cleaner) to back flush the heater core. To know if the heater core is plugged up, disconnect the 2 heater hoses at the heater core and somehow connect a hose to one end of the core. Try to flow water through, it doesn't have to be high pressure, and there should be a steady stream. If not, try to get the cleaner directly into the heater core and let it soak. Then back flush it and a lot of scale should be seen coming out.

2) The other problem is that since the heater core is the highest point in the cooling system air tends to get trapped in it. This should only happen if the coolant goes low or someone has opened the system to service it. There are a number of ways to bleed the system (vacuum bleed, pressure bleed). One of the hoses should have a small hole in it for bleeding air, pull that hose back just enough so the hole allows coolant to flow out of it. Then accelerate the engine and get a good flow coming out, when no more air bubbles are present, there should be good heat.

3)I've heard of bad water pumps with low flow causing issues as well as an aux. water pump not fuctioning correctly leading to poor heat output from the heater core. Someone should confirm this as I cannot get much information on the aux water pump(electronic pump by the alternator that is supposed to run after the car is shut off).

Hope this helps.