Magento – Product List Widget

The new products widget does not support filtering by category.
but you can create your own block [module]/product_new that extends Mage_Catalog_Block_Product_New.
the only thing you need to add in your block are these methods.

//category member var
protected $_category = null;
//load the category
public function getCategory() {
    if (is_null($this->_category)) {
        if ($this->hasData('category_id')) {
            $category = Mage::getModel('catalog/category')->setStoreId(Mage::app()->getStore()->getId())->load($this->getData('category_id'));
            if ($category->getId()) {
                 $this->_category = $category;
            }
        }
        //if the category is not valid set it to false to avoid loading it again.
        if (is_null($this->_category)) {
            $this->_category = false;
        }
    }
    return $this->_category;
}
//override the _beforeToHtml() method
protected function _beforeToHtml() {
    parent::_beforeToHtml();
    //if a category is specified filter by it.
    if ($this->getCategory()) {
        $this->getProductCollection()->addCategoryFilter($this->getCategory());
    }
    return $this;
}
//Change the cache key so you won't get the same products for different categories when the cache is on
public function getCacheKeyInfo()
{
    $cacheKeyInfo = parent::getCacheKeyInfo();
    if ($this->getCategory()) {
         $cacheKeyInfo[] = $this->getCategory()->getId();
    }
    else {
         $cacheKeyInfo[] = 0;
    }

    return $cacheKeyInfo;
}

Now all you need to do is use your block in the {{block}} directive

{{widget type="[module]/product_new" category_id="12" products_count="4" template="catalog/product/widget/new/content/new_grid.phtml"}}

while you using this code

{{widget type="Magento\Reports\Block\Product\Widget\Viewed" page_size="5" template="widget/viewed/content/viewed_grid.phtml"}}

or insert widget in static block with code you should in Cache Management page disable FPC then reload home page (ctrl + f5).