My york furnace works fine other than the blower turning on about 20 seconds after
Ignition. It blows cold until heat exchanger heats up. I have had control board and blower motor changed, at seperat times, and niether technician could say why this was happening other than it's a york. What could be some of the reasons that this is happening? Thank You.

That is about the time mine takes. Where the ductwork is located may be a clue. In our old house, the supply ductwork was in an unheated garage and always blew cold air at startup, even with the insulated ductwork. New house, the ductwork is in a partially conditioned space. It also depends on ambient temperature, particularly if ductwork in in an attic and the temp is 10*. Older units had an adjustable fan ON switch, most new ones are not adjustable. Search "YORK Model.....Fan on settings".

How long before your air is warm, is the question for me?
My Bryant isn't very old and starts blowing cold air for maybe 15 seconds. I don't consider that bad - I'd rather have that than the first warm air not getting to the vents.

I agree. It appears that 20 seconds after ignition the blower comes on is the way it's designed according to the manual.

The limit switch never reaches the fan on setting, so it blows ice cold air threw the house for a good 5+ minutes before you feel any heat.

"The limit switch never reaches the fan on setting".
Well then, how does the fan come on? There is a Fan ON and Fan OFF switch, likely not adjustable, and a high limit switch. I just can't picture the heater running, gas burners on, taking 5 minutes to get heat. Have you looked at the ductwork? Maybe disconnected? Watch the burners and tell us what happens after they come on.
https://i.ebayimg.com/thumbs/images/g/g14AAOSwHaBWiraG/s-l225.jpg (https://www.ebay.com/itm/York-Coleman-Luxaire-Furnace-S1-02541318000-Rollout-Limit-Switch-Manual-Reset/112698299728?_trkparms=aid%3D222007%26algo%3DSIM.M BE%26ao%3D1%26asc%3D20150313114020%26meid%3D23232a 9c0a0a4c1f9290984feede8ada%26pid%3D100338%26rk%3D2 0%26rkt%3D30%26mehot%3Dlo%26sd%3D112718066622&_trksid=p2141725.c100338.m3726)
Most fan switches today look like this and are not adjustable. The high limit switch looks similar with a red button.

All the duct work is connected properly. The burner lights and stays lite. The limit switch is mechanical dail, not electronic. The dial has three settings, Fan-off, Fan-on and upper limit. The settings on the dial are as follows, Fan-off is 80 degrees, Fan-on is 110 degrees and upper limit is 160 degrees. Once the furnace lights, it is only 7 seconds (this time I timed it) and the blower turns on. The dial never moved and was bottomed out at 50 degrees. Cold air blows threw all registers in the house for over 5 minutes before you feel any heat at all from any register. There is no zoning. The air flow thew the system is great, trust me, the cold air flow, before heat is felt, is AWSOME!

Time for the model numbers for specific trouble shooting research. Do you have an owners manual? Where is your furnace located? What is the ambient temperature of the air that's drawn in through the circulating fan? They recommend at least 55* but we all know that garages and attics can fall far below that during the winter. The model I researched had a blower and thermostat adjustment section so can we have that specific model number?

Was up in the attic putting away XMAS stuff and decided to check the fan on time. Amana/ Goodman 80%. No fan control switch, push button reset high limit. Using my counting, 23-25 seconds for fan ON after burner ignition. Counted 3 times. Ambient temperature was28*. I think you have other issues rather than a heater problem. All ductwork insulated? What is your outside temp?