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Hi Chris -
The problem appears to be number of marks you're drawing. In the attached workbook, you're asking Tableau to render 1.98M distinct marks. Each mark takes time and RAM to draw - in your case a grand total of 800-900 MB on first render. I saw Tableau use 1.4 GB of RAM when I made a single modification to your base viz (removing "Label" from Level of Detail). Based on the amount of RAM utilized when I made a single change, it's not surprising you're seeing OOM issues.
Do you need to draw both sets of polygons out of the box? My removing SubpolygonID from Level of Detail, we "only" drew 1.02 M marks and needed 500 MB of RAM.
I'm not a mapping expert, but I wonder if you could smooth/blend these polygons somehow so that fewer points need to be drawn on a per-polygon basis - your maps might lose a touch of fidelity, but you'd be able to do more before you run out of runway.
You might also consider applying a filter so that we only show (therefore draw) polygons associated with a distinct region of the UK rather than the whole country?
Essentially I do not need to see the polygons with such fine detail, I am just unsure to go about the 'smoothing' of my initial data which is contained within an Access database. I do however need to see an overview of the entire country. I will see what I can find with reducing the number of rows in my data.