How do I Bind to elements in a Window from an element in a UserControl that has been added to the window

You can't do this directly.
The Window and SecondControl are two different contexts (WPF XAML NameScope) and the ElementName is limited by NameScope.
"You can refer to elements in code only if they are registered to the appropriate NameScope"
MSDN: Binding.ElementName[^]

So, I see two possibilities:
1. If the FirstNameTextBlock/LastNameTextBlock controls of the Window are NOT "hardcoded", but are instead data bound to the Window's DataContext, then the SecondControl can use the same binding descriptions, since it has inherited the DataContext from the Window:

<Window x:Class="WpfApplication1.MainWindow"
        xmlns:local="clr-namespace:WpfApplication1">
    <Grid x:Name="LayoutRootGrid">
        <Grid.RowDefinitions>
            <RowDefinition Height="100"/>
            <RowDefinition Height="100"/>
            <RowDefinition Height="*"/>
        </Grid.RowDefinitions>
        <TextBlock x:Name="FirstNameTextBlock" Grid.Row="0"
                 Text="{Binding Path=FirstName}"/>
        <TextBox x:Name="LastNameTextBlock" Grid.Row="1"
                 Text="{Binding Path=LastName}"/>
        <local:SecondControl x:Name="SecondControl" Grid.Row="2"/>
    </Grid>
</Window>

<UserControl x:Class="WpfApplication1.SecondControl">
    <StackPanel Background="CadetBlue">
        <TextBlock HorizontalAlignment="Left" VerticalAlignment="Top"
                   Text="{Binding Path=FirstName}" />
        <TextBlock HorizontalAlignment="Left" VerticalAlignment="Top"
                   Text="{Binding Path=LastName}" />
    </StackPanel>
</UserControl>

2. (and probably better) would be to have the SecondControl contain DependencyProperties that can take the binding where the SecondControl is used in the Window:

<Window x:Class="WpfApplication1.MainWindow"
        xmlns:local="clr-namespace:WpfApplication1">
    <Grid x:Name="LayoutRootGrid">
        <Grid.RowDefinitions>
            <RowDefinition Height="100"/>
            <RowDefinition Height="100"/>
            <RowDefinition Height="*"/>
        </Grid.RowDefinitions>
        <TextBlock x:Name="FirstNameTextBlock" Grid.Row="0"
                 Text="John"/>
        <TextBox x:Name="LastNameTextBlock" Grid.Row="1"
                 Text="Smith"/>
        <local:SecondControl x:Name="SecondControl" Grid.Row="2"
                             FirstName="{Binding ElementName=FirstNameTextBlock, Path=Text}"
                             LastName="{Binding ElementName=LastNameTextBlock, Path=Text}" />
    </Grid>
</Window>

<UserControl x:Class="WpfApplication1.SecondControl" x:Name="SecondControl" >
    <StackPanel Background="CadetBlue">
        <TextBlock HorizontalAlignment="Left" VerticalAlignment="Top"
                   Text="{Binding ElementName=SecondControl, Path=FirstName}" />
        <TextBlock HorizontalAlignment="Left" VerticalAlignment="Top"
                   Text="{Binding ElementName=SecondControl, Path=LastName}" />
    </StackPanel>
</UserControl>

In SecondControl.xaml.cs:

public static readonly DependencyProperty FirstNameProperty =
    DependencyProperty.Register("FirstName", typeof(string), typeof(SecondControl), new UIPropertyMetadata(string.Empty));

public string FirstName
{
  get { return (string)GetValue(FirstNameProperty); }
  set { SetValue(FirstNameProperty, value); }
}

public static readonly DependencyProperty LastNameProperty =
    DependencyProperty.Register("LastName", typeof(string), typeof(SecondControl), new UIPropertyMetadata(string.Empty));

public string LastName
{
  get { return (string)GetValue(LastNameProperty); }
  set { SetValue(LastNameProperty, value); }
}