Hi... First of all, you need to have a general knowledge about your vehicle. Checking and finding the problem anyway is good.

Al Nael is the sales & service agent here.Please understand, they will not be correct always. Who ever goes to the company for any small issue related with transmission issue, they say, its BTRA gearbox, the life of the gearbox is 100'000Kms. Next, you ned to redo the gearbox!!! and cost 7000Qrs.

My friends car had a sudden stop, the radiator hose broken and he couldnt guess even whats the issue is. When he took the car to al nael, they start the engine, the sound was lil noicy beacsue of no water in the engine. They told simply, engine burnt out du to less water. What to do next??

My friend replied, replace the broken and old hoses connecting radiator. they did. Now no issues. And he was there untill the works done. So dont believe them blindly.

Coming to your issue, the delayed engagement, 1. check the TCU has any rusty connections. 2. Check the oil level and be prsent at the spot. All technicians are like robos, not as much knowlegde on how much oil to fill and how to. More than that, its not their vehicle. What ever they do, they will get salary. So get some knowledge about your gear box first.
3. Personally check any leaks on the gearbox by seeing under the ramp. 4. Any warning signs on the display?

Before starting your vehicle in the morning, move the gear lever from P-1 and couple of times. Then Start. within a minute, the gear will engage. this is because of less fuel.

NOTE: THE ATF MUST BE USED FROM THE AL NAEL. ITs DIFFERENT FROM THE OTHER GEAR OIL PRODUCTS. ANY OTHER OILS WILL SPOIL YOUR GEARBOX AND MAKE DELAYED TRANSMISSION. THIS PER LITER IS 69Qrs. You may fill atleast 6 liters in the gearbox.