Thank you DrAzzy. After struggling for 3 days I finally managed to update it! So I will describe how I have done it in the lines below:
The first thing that I noticed was that I always had 8 successful passes and 0 fails. After checking out a lot of websites/forums I noticed that all the people that successfully updated the shield received 11 successful passes and 0 fails and maybe 1 warning - in comparison with 8 passes and 0 warnings of mine.
The 2nd thing that I noticed was that there were a lot of people talking about bright blue led turning on after loading first file "wifi_dnld.elf" - and that in my case wasn't true. There was no blue led turned on.
So here is the solution that worked for me:
I put both files wifi_dnld.elf and wifiHD.elf into the same easily accessible map. (C:\temp).
Next, I downloaded FLIP and updated its drivers - it is a well-known procedure so just google it.
Now, here is the tricky part. What I did was that I plugged my wifi shield (with no jumper on-because I simply forgot) over the cable with my PC and after navigating at the CMD to C/Program Files/Atmel/Flip 3.4.7/bin/ I executed the whole command:
batchisp.exe -device AT32UC3A1256 -hardware usb -operation erase f memory flash blankcheck loadbuffer "C:\temp\wifi_dnld.elf" program verify start reset 0
There weren't any "enters" or anything like that in the command and after loading the first command I saw the blue led shining. And also there were 11 passes - normally were 8. So, now I unplugged my shield from the PC. Took off the jumper and plugged it back in. After that I used the same command but for different file: batchisp.exe -device AT32UC3A1256 -hardware usb -operation erase f memory flash blankcheck loadbuffer "C:\temp\wifiHD.elf" program verify start reset 0
And that was it. I unplugged the jumper, put the board on uno and here it is! Working and updated version of Firmware. I can even use Blynk with it now.
I hope this saves some nerves and time in the future for someone.
Kind regards, Greg