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So, I bled the brakes. It was the first time I'd done anything like that and to be honest, it was relatively straight forward. (Thanks @snottyotter for recommending the kit) I think there's still a tiny bit of air is the system, as the brakes aren't quite as stiff as I'd prefer, but they're more than adequate! It took about an hour to bleed the rear, as I it took ages to find out how to open the nipple (there's an allen key port above) and all the instructions I could find mentioned I needed to use a 7mm spanner. But once this was figured out it was all fine. The front brake took about 15mins to do.
All Shimano use mineral oil, which is all broadly the same with maybe some different additives. One thing you can't mix it with is DOT fluid which is used in SRAM and hope etc. You can't use DOT in shimano systems, DOT is usually clear/slightly yellow but I'm sure some mineral oils are too.