if it is a first floor flat, then the draught is probably coming from gaps in the brickwork under the floor, as plastered celings are not usually draughty unless they have holes in them. If the house was built since about 1940 then it probably has cavity walls and the joists will protrude into the cavity, usually with gaps round them. As you don't know if it a Victorian house or a 1990's house I can't guess.
you could probably deal with it best by taking up the floorboards adjacent to the wall, and stuffing loft insulation between the joists. If you see any visible gaps you could fill them with mortar or expanding foam.
If you are troubled by dust you can hoover out the space under the floor while you have some boards up. A hoover hose will be useful as you can poke it along.
You can use an acrylic or silicon flexible filler in the gap between the skirting and the floor, but do any floorboard work first, and clean the gap well or it will not stick. You will have to cut it out if you want to move the floorboards later.
Look out for loft insulation where the fibres have a coating to prevent glass dust and shed fibres. It is some of the Knauf grades, it will mention "Ecose" on the packaging and is widely sold. Not the yellow stuff.
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