Bootstrap contact form not sending emails (FormTools neither) - php

Question

I've tried searching high and low for a resolution to get a contact form to send e-mails for a few days now. Currently using Bootstrap. My host allows you to install FormTools but, although the enquirys are visible within FormTools, e-mails still do not arrive.
<form action="contact.php" method="post">
<div class="row">
<div class="form-group col-md-6">
<label for="exampleInputName">Full name</label>
<input type="name" name="cf_name" class="form-control" id="exampleInputName1" aria-describedby="nameHelp" placeholder="Eg. John Smith">
</div>
<div class="form-group col-md-6">
<label for="exampleInputEmail1">Email address</label>
<input type="email" name="cf_email" class="form-control" id="exampleInputEmail1" aria-describedby="emailHelp" placeholder="example#example.co.uk">
</div>
<div class="form-group">
<label for="exampleSelect1">Subject of enquiry</label>
<select class="form-control" name="cf_subject" id="exampleSelect1">
<option>CCTV & Security</option>
<option>Door Access</option>
<option>Electrical</option>
<option>Networking</option>
<option>PC Repairs</option>
<option>Smart Thermostats</option>
<option>Other</option>
</select>
</div>
<div class="form-group">
<label for="exampleTextarea">Message contents</label>
<textarea class="form-control" name="cf_message" id="exampleTextarea" rows="3" placeholder="My enquiry..."></textarea>
</div>
<button type="submit" class="btn btn-primary">Submit</button>
</div>
</form>
contact.php:
<?php
$field_name = $_POST['cf_name'];
$field_email = $_POST['cf_email'];
$field_subject = $_POST['cf_subject'];
$field_message = $_POST['cf_message'];
$mail_to = 'myemail#live.com';
$subject = 'Message from a site visitor '.$field_name;
$body_message .= 'From: '.$field_name."\n";
$body_message .= 'Subject: '.$field_subject."\n";
$body_message .= 'E-mail: '.$field_email."\n";
$body_message .= 'Message: '.$field_message;
$headers = 'From: '.$field_email."\r\n";
$headers .= 'Reply-To: '.$field_email."\r\n";
$mail_status = mail($mail_to, $subject, $body_message, $headers);
if ($mail_status) { ?>
<script language="javascript" type="text/javascript">
alert('Thank you for the message. We will contact you shortly if required.');
window.location = 'contact.html';
</script>
<?php
}
else { ?>
<script language="javascript" type="text/javascript">
alert('Message failed. Please, send an email to myemail#live.com');
window.location = 'contact.html';
</script>
<?php
}
?>
I've not had much experience with PHP and I've tried a few samples of code but I feel like I'm making no progress and although I've tried searching, I'm unsure how to implement certain solutions.
If someone could help or guide me (even point out a mistake I've probably made myself) that would be amazing.
Thanks for your help!