I note you are in the UK; from mid month, you can apply a Lawn Feed, either a granular one or a liquid mixed in a can. Some of the granular ones also contain moss killer and weed killer - if your lawn is largely weed free, I suggest you avoid one with weedkiller in it. This is because the weedkiller inevitably 'checks' lawn growth, and they are not terribly effective anyway when in a combined formulation. Separate lawn weedkillers such as the Weedol one are more effective if you need weed control. You can reapply the feed usually six weeks after the first application, but it's easier to get the dosing right if you use a lawn spreader designed for the purpose, they're not usually very expensive.
For granular feeds, the grass should not have been cut for at least 4 days, and you need a dry, wind free or light winds only day in order to spread it - you need the blades of grass to be dry when you apply it. If it doesn't rain after application within 48 hours, then you will need to put a sprinkler on it to wash it into the soil. Liquid feeds have different rules - check the products at the garden centre and follow any instructions on the package, whether you choose a granular or liquid formulation.
As for seeding, this can only be done when chemicals have not been used for the previous 6 weeks, and no chemicals should be applied for another six weeks afterwards. From the looks of it, not sure you need to overseed - if you use lawn fertiliser and proper lawn care such as cutting regularly, but not too short when you do, it'll probably grow rapidly and won't look 'patchy'. Now all you need is for it to stop raining....and finally, a question - how sharp are the blades on your mower? I ask because in closeup, your lawn looks like it has slightly fuzzy or chewed ends where it's been cut, though this could have been because it was pretty wet when you did cut it...
