Dimikius,
<!DOCTYPE html>
<html>
<head>
<title>Untitled</title>
<meta charset="utf-8">
<style type="text/css">
.overlay-form{
display: none;
min-width: 100px;
min-height: 100px;
border: 1px solid red;
position: absolute;
top: 50%;
left: 50%;
transform: translate(-50%, -50%);
}
</style>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.4/jquery.min.js"></script>
<script>
$(function() {
$("tr").each(function(indx, tr){
$(".schedule__apply",tr).click(function( event ) {
$(".overlay-form").show().find('[name="data_start"]').val($(":first",tr).text())
});
});
});
</script>
</head>
<body>
<table>
<tr>
<td id="start_1">12.09.2017</td>
<td>
<button id="button_apply_1" class="schedule__apply">Записаться
</button>
</td>
</tr>
<tr>
<td id="start_2">13.09.2017</td>
<td>
<button id="button_apply_2" class="schedule__apply">Записаться
</button>
</td>
</tr>
</table>
<form class="overlay-form">
<input type="hidden" name="data_start" value="">
<input name="name" placeholder="Ф.И.О." required="true" type="text"/>
<input name="tel" type="tel" placeholder="ТЕЛЕФОН" required="true"/>
<button class="overlay-form__apply">Записаться</button>
</form>
</body>
</html>