Hello Ruth
There is no need to loosen the turf as grass roots and worm channels have already done that.
All the composts you mention are suitable and worms should enjoy them and gradually integrate them with the soil below. Straw is not a good plan as it needs nutrients to decompose and will use some from the composts around it if you put it in the bed. I use straw for the compost heap. Wood ash is alright but not too much, say a bucket per bed of 8×4 feet. Put the rest in your compost heap, little and often.
Compost with weed seeds could be put on as first layer, and your nice worm compost as surface layer, then cardboard on top. The latter may need replenishing with another layer of card on top if it is rotting and starting to blow away by late winter.
Then by April I would remove any remaining cardboard and you can start sowing and planting. Weeds from your compost will need pulling or hoeing but will be easy to deal with (when small, as seedlings) because compost is so soft and friable. Once the first flush of weed seeds have germinated, weeding becomes much less and easier, but you must do it regularly, even just for a minute a week, or whatever it takes to keep the bed clean and not have any more weeds going to seed.
The cardboard layer is a one-off and does not need repeating. It may even be unnecessary if your compost layer on the grass is six inches deep, which is thick enough to exclude all light and prevent re-growth of most perennial weeds such as buttercup.