What I think happens is that in the email send function Mage_Customer_Model_Customer::sendPasswordResetConfirmationEmail
the store id attached to the loaded user is used and not the current store you are on.
Open this file app/code/core/Mage/Customer/Model/Customer.php
Find below code on same file.
public function sendPasswordResetConfirmationEmail()
{
$storeId = $this->getStoreId();
if (!$storeId) {
$storeId = $this->_getWebsiteStoreId();
}
$this->_sendEmailTemplate(self::XML_PATH_FORGOT_EMAIL_TEMPLATE, self::XML_PATH_FORGOT_EMAIL_IDENTITY,
array('customer' => $this), $storeId);
return $this;
}
Replace with below code.
public function sendPasswordResetConfirmationEmail()
{
$storeId = $this->getStoreId();
if (!$storeId) {
$storeId = Mage::app()->getStore()->getId();
}
$this->_sendEmailTemplate(self::XML_PATH_FORGOT_EMAIL_TEMPLATE, ;
array('customer' => $this), $storeId);
return $this;
}
For best practice,don't change core file..Put same file inside local directory like this:- app/code/local/Mage/Customer/Model/Customer.php
Other Solution
Please configure both store different forget password email template,Kindly review below URLs:-
https://www.templatemonster.com/help/magento-how-to-edit-store-emails.html#gref
https://haijerome.wordpress.com/2010/11/24/magento-email-templates/
Hope it's helpful for your.