Where do I begin?

Quite simply, how do I sort this mess of a lawn? It’s a bumpy lawn and I don’t think the hot weather has helped much, along with the tree most likely stripping nutrients that the grass requires!

There are also some dog wee burn areas

 

I'd be tempted to give it a good raking or scarifying and then scatter lawn seed and tread or roller the seed in. Then keep moist for a few weeks until the new grass gets established. I'd also give the lawn a good feed in the autumn and then again in spring.

 

My lawn has suffered like this in the drought we’ve had; over two months of no rain. The finer grasses, the fescues, have really taken the brunt, which may be the case with yours. Mine is recovering but it is taking time. There is no real harm in scarifying but do look closely for fine new growth before you do so. I think when autumn comes it will recover.