Mapbox: show only one country on base map?
Unfortunately it is not possible to filter by a certain geography when selecting the data source for a layer. If you're working with Mapbox's tiles, they'll always cover the entire planet.
There is the possibility to restrict the map to a certain (rectangular) bounds, with the map.setMaxBounds method (https://www.mapbox.com/mapbox-gl-js/api/#Map#setMaxBounds). This might work reasonably well since you want to restrict the view to the UK, but might not be suitable depending on the geometry you want to restrict the view to.
As a workaround you could create a dataset and add a polygon that covers the entire planet except for the extent you want to show in your map. Then add this dataset in your style as the top most layer and style it with whatever you'd like the empty space in your map to look like.
The workaround has been also suggested in this SO answer, if you can get maptiles for the UK only (the SO answer gives a link, but for Germany) you can upload them to Mapbox as a dataset, export it to tilesets, and then to a map as a layer. Delete all other layers
You have also the possibility to Style a single country in Mapbox studio ref this tutorial. The other countries are still shown, but you can style your map in a way to highlight UK
There is a option that allows you to show only one country highlighted. But drawback is, you lose all the layers and tile-level details. Here is the link https://www.mapbox.com/videos/how-to/map-a-single-country-in-mapbox-studio/.
But if you want to include the tile details as well as whole world map but in that map only one or two or custom amount of countries highlighted, then, from above link of Natural Earth download the zipped file of ne_10m_admin_0_countries and upload.
Go back to your map style, and instead of making a blank one. Add another layer of ne_10m_admin_0_countries in your current map. Select the country in the filter option (in same way as shown in video). And change its opacity. That's it.
Hope it helps. Have a great day.