Looks like speedwell to me. It is very difficult to contol and I know of no really effective way to do it from the conditions which you have described. Perhaps other members will have ideas on how to stop speedwell which has conquered a lawn and is now on the march elsewhere.

Mostly liverwort, you will need a different chemical on the slabs from the one on the lawn.

yep, looks like liverwort and moss rather than speedwell. It'll be in the lawn and here because of the damp and reasonably mild winter conditions. A combined mosskiller/weed/feed treatment for lawns should sort out the liverwort as well as the moss (but check the information on the pack). Aerating the lawn will help . Ensure any treatment you use which kills mosses/algae/liverworts is suitable for use on paving, as Jimmy implies.

Ok. Thanks everyone. I'll pass on the good news! ?

Looks like Marchantia polymorpha, Common Liverwort.