Bird of prey identification help please! Merlin?

Hi! New here so I hope I'm posting in the right place. 

We have recently been hearing a bird call during the day, it makes a repetitive one note chip or kuk sound, not sure how to explain it. It makes the sound about 10 times then stops, then does it again. I've been going out of my mind trying to identify it. I eventually settled on a kestrel like sound, but "my" birds called is a bit slower paced. This search came up with a merlin? 

I FINALLY managed to spot the bird in question yesterday!! 

I shall attach pictures. It was grey in colour (from what I could see from afar), it had a relatively long tail and was about the size of a magpie (I think)... it's hard to remember from my one sighting. We have parakeets nesting around us, and the shape of the bird in flight reminded me of them, but bigger. (It flew from the top of one tree to another). 

Any help would be appreciated. I'm driving myself mad trying to identify my new neighbour! 

Also to add, I'm living in an urban area in the UK, Midlands.