You may well be blessed with ground elder but the top piccie is a different weed - member of the buttercup family I think.
Re weedkiller, if you intend to be completely organic from day 1 then obviously it's a no-go.
Assuming you're happy to use it, and you intend to use a glyphosate based one, you might get away with using it at this time of the year but it works best when the plant is actively growing and thus can take the chemicals down to the roots.
Apply it when the foliage is dry, and when it will remain dry for at least the next 6 hours, and when there's no chance of wind drifting the spray anywhere else. Do not wash it off, and do not cover the plant for a couple of weeks, until the foliage starts turning yellow.
TBH, I would not rely on weedkiller at this time of year but would just try to dig out what I can over the next few months. (I wouldn't leave starting clearance until the spring as time will in all likelihood overtake you.) I would then use weedkiller on active new growth in the spring. But others may well disagree.
You could cover weeds over the winter but I don't think they will be growing much; it will deprive them of light but won't kill them off, and will provide a haven for slugs. However, do be aware in the spring that they will start to regrow as soon as your back is turned so do be prepared to cover that which you haven't dug out or weedkilled, or it will get away from you.
I would check whether the grass is indeed couch grass or ordinary grass. The roots will be the giveaway - piano wire is bad, fibrous is good. If good, you can lift the turves and stack them upside down in a pile in order to rot down and provide compost in a year or so's time.