Sky Q Minis are trouble some on a mesh system, the Qs need a direct link back to the box or to go through a sky booster to work well, I have a mesh system and the Qs are the only thing that don't work properly on this as the Qs are on a different switch to the main box.
First call Sky, tell them your system is not getting Wifi reliably to the mini, they might send you a booster free of charge, though as you are on BT they might not, you can fit midway between the main q and mini q to boost wifi, you need to get 20-30Mb to the box reliably. once you have the sky box working this way you might not need your extenders as the mini will give coverage.
If that doesn't work and you can't wire then Powerline is a reliable way to fake a direct link to a mini, I actually do this, to get this to work you have to connect to ethernet of powerline and mini, reset the network on the mini and do not press WPS to link mini Qs, once connected go into engineers menu on the mini and turn off wifi, downside to that approach is that the Minis are actually a Wifi Mesh system (when you have Sky BB at least, not 100% sure that holds true if you are with someone else ) and unknown to you, you might actually be relying on it for coverage in your house, so once this is down you might have buggerall wifi where the mini is.
A Mesh wifi system will give you great wifi around the house and you won't need extenders that are often poor for wifi performance in general but sky help forum is full of people who can't make this work well on minis without down grading one of the mesh satellite to be a simple AP which is not desirable if you want mesh for wifi coverage.
The fact you are using extenders suggests you don't have good wifi and this is the main thing you need to fix, this could just be the case of getting a good wifi or mesh router but if you have to access servers remotely then you will have double NAT to deal with due to BT modem not havng a modem only mode.
