I confirm it has nothing to do with your theme. I present here the solution so it is available for others. The code has to be placed in functions.php:
function populate_specific_checkout_field($input, $key ) {
global $current_user;
switch( $key ) {
case 'billing_first_name':
case 'shipping_first_name':
// return the first name
return $current_user->first_name;
case 'billing_last_name':
case 'shipping_last_name':
// return the last name
return $current_user->last_name;
}
}
add_filter('woocommerce_checkout_get_value', 'populate_specific_checkout_field', 15, 2 );