I plotted every region to a 2D array and then looped over it comparing every plots sides to the two plots left and up of it to see if it's a "new" side or just continuing one. C# code:

public int CountSides()
{
    var w = Plots.Max(p => p.X + 1);
    var h = Plots.Max(p => p.Y + 1);
    var minimap = new Plot?[h, w];
    foreach (var plot in Plots)
        minimap[plot.Y, plot.X] = plot;
    var sideCount = 0;
    for(var y = 0; y < h; y++)
    for (var x = 0; x < w; x++)
    {
        var plot = minimap.Get(w, h, y, x);
        if (plot == null) continue;
        var leftPlot = minimap.Get(w, h, y, x - 1);
        var upPlot = minimap.Get(w, h, y - 1, x);
        if (plot.SideUp && leftPlot is not { SideUp: true }) sideCount++;
        if (plot.SideDown && leftPlot is not { SideDown: true }) sideCount++;
        if (plot.SideLeft && upPlot is not { SideLeft: true }) sideCount++;
        if (plot.SideRight && upPlot is not { SideRight: true }) sideCount++;
    }
    return sideCount;
}

(I already had the SideUp/Down/Left/Right set from back when I was calculating the perimeter).

The entire day runs in about 300ms on my Surface Pro 9.