From the images, it appears your mower may need its blades sharpening - I can see chewed up ends left behind rather than cleanly cut ones, though that may just be the result of cutting overly long grass with a cylinder mower. The images do suggest you had not cut the grass before May, possibly late into May and maybe not at all this year till May; if that's the case, then a brown, uneven appearance is inevitable if you left it that long, especially if you cut it too short.
Regardless, it will recover over time; do not apply fertilisers or mosskillers while we are in drought. What it needs is water, preferably in the form of good rainfall. Looking at the weather forecast, there does seem to be a possibility of rain within the next few days, and where you live is, in theory, a higher rainfall area of the UK. It's difficult to say if there is any moss present because of the lawn's current appearance - when the weather is damp and it starts growing again, it should be easier to tell then.
In respect of 'No Mow May', I don't know who thought that one up, but I don't think it was a good idea at all. Late April and May are when grass in the UK grows really, really fast; not cutting for the whole of May means you're likely to end up with grass 4 feet high, not something than can easily be cut with an electric mower (and particularly not with a cylinder mower) because it will burn the motor out, you'd need to use a strimmer or scythe first. The most sensible idea is to cut (probably once a week) to leave it high during May (and ongoing if we are in drought) which is what you were attempting in the first place.