During crassing a border I stated my car as goods to follow and I have Canadian customs paper that this car is my goods to follow.
My car was in the USA and during shipping and custom clearence to Canada my transport company paid custom tax (6.1%), GST (5%) and HST (8%).

Can I and how I can apply to tax refund? Which taxes I can expect to get back?

Contact CRA for a form (maybe able to download it), or the broker that cleared your shipment. Fill it in and send it off to CRA with a copy of your goods to follow list and irrefutable proof that this is the car on your list. They will then decide if and what taxes you get back.

Has anybody tried to do this? And how successful?

Did you come from the UK to Canada? Was the car a new car?

No, I came from Armenia. Car was used (2004 Toyota 4Runner) and I bought it before in the USA (New York).

How long had you owned it before coming to Canada?

2 months

If you were legally resident in the US, had an address and used the vehicle, you might get them back if you met the following criteria:

Ownership, possession, and use requirements

For your goods to qualify for duty- and tax-free importation as settler’s effects, you must have owned, possessed, and used them before your arrival in Canada. If you have the sales receipts and registration documents for these items, you can use them to help prove they meet these three requirements.

It is important that you meet the three requirements of ownership, possession, and use. For example, if you owned and possessed the goods without using them, the goods are subject to duties.

Based on previous posts that was never the case. He, or she, picked up the car on the way to Canada.

Verbatim from CRA website. Now of course a lot depends on what you tell them, however falsely signing a declaration is not a good idea.

If you've had it less than 6 months it doesn't count as your goods to follow and isn't exempt from duty.