Per my testing, this issue is caused by the change event in .SPServices.SPCascadeDropdowns will not be triggered because the selected option value is not changed.

To resolve this, we need to overwrite the autocomplete source code and cascading function.
Custom cascading function -- refer to:
http://www.markrackley.net/2014/05/20/cascading-drop-down-lists-in-sharepoint-office-365-using-rest/
Also call the cascading function in this._on
method within autocomplete function. After that, when the parent value is changed, the child values will be changed.
Demo:
<script src="https://code.jquery.com/jquery-1.11.3.min.js" type="text/javascript"></script>
<script src="https://code.jquery.com/ui/1.11.4/jquery-ui.min.js" type="text/javascript"></script>
<link rel="stylesheet" type="text/css" href="https://code.jquery.com/ui/1.11.4/themes/ui-lightness/jquery-ui.css"/>
<style>
.custom-combobox {
position: relative;
display: inline-block;
}
.custom-combobox-toggle {
position: absolute;
top: 0;
bottom: 0;
margin-left: -1px;
padding: 0;
}
.custom-combobox-input {
margin: 0;
padding: 5px 10px;
}
</style>
<script type="text/javascript">
$(document).ready(function(){
HillbillyCascade({
parentFormField: "Company", //Display name on form of field from parent list
childList: "CostCenter", //List name of child list
childLookupField: "Title", //Internal field name in Child List used in lookup
childFormField: "CostCenter", //Display name on form of the child field
parentFieldInChildList: "Company" //Internal field name in Child List of the parent field
});
(function( $ ) {
$.widget( "custom.combobox", {
_create: function() {
this.wrapper = $( "<span>" )
.addClass( "custom-combobox" )
.insertAfter( this.element );
this.element.hide();
this._createAutocomplete();
this._createShowAllButton();
},
_createAutocomplete: function() {
var selected = this.element.children( ":selected" ),
value = selected.val() ? selected.text() : "";
this.input = $( "<input>" )
.appendTo( this.wrapper )
.val( value )
.attr( "title", "" )
.addClass( "custom-combobox-input ui-widget ui-widget-content ui-state-default ui-corner-left" )
.autocomplete({
delay: 0,
minLength: 0,
source: $.proxy( this, "_source" )
})
.tooltip({
tooltipClass: "ui-state-highlight"
});
this._on( this.input, {
autocompleteselect: function( event, ui ) {
ui.item.option.selected = true;
this._trigger( "select", event, {
item: ui.item.option
});
// trigger cascading here when result is changed
DoHillbillyCascade(ui.item.option.value,{
parentFormField: "Company", //Display name on form of field from parent list
childList: "CostCenter", //List name of child list
childLookupField: "Title", //Internal field name in Child List used in lookup
childFormField: "CostCenter", //Display name on form of the child field
parentFieldInChildList: "Company" //Internal field name in Child List of the parent field
});
},
autocompletechange: "_removeIfInvalid"
});
},
_createShowAllButton: function() {
var input = this.input,
wasOpen = false;
$( "<a>" )
.attr( "tabIndex", -1 )
.attr( "title", "Show All Items" )
.tooltip()
.appendTo( this.wrapper )
.button({
icons: {
primary: "ui-icon-triangle-1-s"
},
text: false
})
.removeClass( "ui-corner-all" )
.addClass( "custom-combobox-toggle ui-corner-right" )
.mousedown(function() {
wasOpen = input.autocomplete( "widget" ).is( ":visible" );
})
.click(function() {
input.focus();
// Close if already visible
if ( wasOpen ) {
return;
}
// Pass empty string as value to search for, displaying all results
input.autocomplete( "search", "" );
});
},
_source: function( request, response ) {
var matcher = new RegExp( $.ui.autocomplete.escapeRegex(request.term), "i" );
response( this.element.children( "option" ).map(function() {
var text = $( this ).text();
if ( this.value && ( !request.term || matcher.test(text) ) )
return {
label: text,
value: text,
option: this
};
}) );
},
_removeIfInvalid: function( event, ui ) {
// Selected an item, nothing to do
if ( ui.item ) {
return;
}
// Search for a match (case-insensitive)
var value = this.input.val(),
valueLowerCase = value.toLowerCase(),
valid = false;
this.element.children( "option" ).each(function() {
if ( $( this ).text().toLowerCase() === valueLowerCase ) {
this.selected = valid = true;
return false;
}
});
// Found a match, nothing to do
if ( valid ) {
return;
}
// Remove invalid value
this.input
.val( "" )
.attr( "title", value + " didn't match any item" )
.tooltip( "open" );
this.element.val( "" );
this._delay(function() {
this.input.tooltip( "close" ).attr( "title", "" );
}, 2500 );
this.input.autocomplete( "instance" ).term = "";
},
_destroy: function() {
this.wrapper.remove();
this.element.show();
}
});
})( jQuery );
$("select[title='Company']").combobox();
$("select[title='CostCenter']").combobox();
// $("select[title='Company']").parent().find("input").change(function(){alert("work");});
function HillbillyCascade(params)
{
var parent = $("select[Title='"+params.parentFormField+"'], select[Title='"+
params.parentFormField+" Required Field']");
var currentParent = $(parent).val();
DoHillbillyCascade(currentParent,params);
}
function DoHillbillyCascade(parentID,params)
{
var child = $("select[Title='"+params.childFormField+"'], select[Title='"+
params.childFormField+" Required Field']," +
"select[Title='"+params.childFormField+" possible values']");
$(child).empty();
var options = "";
var call = $.ajax({
url: _spPageContextInfo.webAbsoluteUrl + "/_api/Web/Lists/GetByTitle('"+params.childList+
"')/items?$select=Id,"+params.childLookupField+","+params.parentFieldInChildList+
"/Id&$expand="+params.parentFieldInChildList+"/Id&$filter="+params.parentFieldInChildList+
"/Id eq "+ parentID,
type: "GET",
dataType: "json",
headers: {
Accept: "application/json;odata=verbose"
}
});
call.done(function (data,textStatus, jqXHR){
for (index in data.d.results)
{
options += "<option value='"+ data.d.results[index].Id +"'>"+
data.d.results[index][params.childLookupField]+"</option>";
}
$(child).append(options);
});
call.fail(function (jqXHR,textStatus,errorThrown){
alert("Error retrieving information from list: " + params.childList + jqXHR.responseText);
$(child).append(options);
});
}
});
</script>
Result:

