The Unifi UAP-AC-M is a really good piece of equipment for this scenario, certainly. If it's externally mounted it will probably provide excellent coverage inside the summer house and in the garden too. I've recently used one just like that.
While configuring Unifi equipment isn't rocket science, it requires a better technical understanding than the average consumer access point - because it isn't a consumer product. It's not really something many traditional electricians are going to be able to configure easily - or at all. What you don't want to do is have someone who charges you by the hour, and takes a long time because they're clueless and take ages to get it working, and even then not properly.
Advice to the OP - the Unifi UAP-AC-M is a really excellent choice for this use but either get someone who has Unifi experience to set it up or take the time to learn how to do it yourself. You can practise configuring it all inside the house before it's set up for the summer house.
If it's mounted on the outside of the summer house, it will need a short length of electrically shielded external grade ethernet cable to connect it to the inside of the summer house. Regular PVC jacketed ethernet cable will quickly degrade in sunshine. You can buy external ethernet cable by the metre from Broadbandbuyer or MS Distribution under the name Tough Cable Pro. It also must have shielded RJ45 connectors and the shield must be earthed - either through the supplied power supply or better still via a separate ethernet surge protector like the Ubiquiti ETH-SP-G2. If the Cat5e cable that runs to the house doesn't have a screen with an earth wire, you will either need to put the mains power supply in the summer house or put an ethernet surge protector there.
Final point about outside mounting: You must configure the access point for specific outdoor use in the Unifi controller.
Good luck with what ever you decide to do and come back with an update or further questions.