.NET Framework 4.5.1 以降でビルドする場合は CurrentMetricUnit と CurrentImperialUnit の set アクセサーは public にしてください

MainWindowViewModel.cs

public class MainWindowViewModel : ViewModel
{
    private double _metricValue, _imperialValue;

    public ICommand ImperialUnitToMetric { get; private set; }
    public ICommand MetricToImperialUnit { get; private set; }

    public MetricUnit CurrentMetricUnit { get; set; }
    public ImperialUnit CurrentImperialUnit { get; set; }

    public double MetricValue
    {
        get { return this._metricValue; }
        set
        {
            this._metricValue = value;
            this.OnPropertyChanged();
        }
    }
    public double ImperialValue
    {
        get { return this._imperialValue; }
        set
        {
            this._imperialValue = value;
            this.OnPropertyChanged();
        }
    }


    public MainWindowViewModel()
    {
        this.CurrentMetricUnit = MetricUnit.Units.First();
        this.CurrentImperialUnit = ImperialUnit.Units.First();
        this.MetricToImperialUnit = new DelegateCommand(
            () => this.ImperialValue = this.CurrentImperialUnit.FromMetricUnit(
                this.CurrentMetricUnit, this.MetricValue));
        this.ImperialUnitToMetric = new DelegateCommand(
            () => this.MetricValue = this.CurrentMetricUnit.FromImperialUnit(
                this.CurrentImperialUnit, this.ImperialValue));
    }
}