SkDemoRecordtTypeSelector.cmp

<aura:component controller="SkRecordTypeSelectionController">
    <aura:attribute name="objType" type="String" required="true" default="Opportunity"/>
    <aura:attribute name="selectedRecordType" type="String"/>
    <aura:attribute name="recordTypeList" type="Object[]"/>
    
    <aura:handler name="init" action="{!c.doInit}" value="{!this}" />
	<div role="dialog" tabindex="-1" class="slds-modal slds-fade-in-open">
        <div class="slds-modal__container">
            <div class="slds-modal__header">
                <lightning:buttonIcon class="slds-modal__close" iconName="utility:close" size="large" 
                                          variant="bare-inverse" alternativeText="Close" onclick="{!c.defaultCloseAction}" /> 
                
                <h2 id="header43" class="slds-text-heading--medium">Please select record Type</h2>
            </div>
            <div class="slds-modal__content slds-p-around--medium">
                <div class="slds-align--absolute-center">
                     <div class="slds-p-top--small slds-p-bottom--small" >
                        <fieldset class="slds-form-element">
                            <div class="slds-form-element__control">
                                <aura:iteration items="{!v.recordTypeList}" var="item">
                                    <label class="slds-radio slds-p-top--small">
                                        <tr>
                                            <td>
                                                <ui:inputRadio name="benefits" change="{!c.onChange}" text="{!item.recordTypeId}" value="true"/>
                                                <span class="slds-radio--faux"></span>
                                            </td>
                                            <td>
                                                <span class="slds-form-element__label">{!item.recordTypeLabel}</span>
                                            </td>
                                        </tr>
                                    </label>
                                
                               </aura:iteration>
                             </div>
                        </fieldset>
                    </div>
                </div>
            </div>
            <div class="slds-modal__footer">
                <lightning:button label="Cancel" variant="neutral"  onclick="{!c.defaultCloseAction}"/>
                <lightning:button label="Continue" variant="brand"  onclick="{!c.onconfirm}"/>
            </div>
        </div>
    </div>
    <div class="slds-backdrop slds-backdrop--open"></div>
</aura:component>

SkDemoRecordtTypeSelectorController.js

({
    doInit : function(component,event,helper){
       // alert('do init get called');
       helper.callToServer(
            component,
            "c.findRecordTypes",
            function(response) {
                //alert(JSON.parse(response));
                var jsonObject=JSON.parse(response);
                component.set('v.recordTypeList',jsonObject);  
                component.set('v.selectedRecordType',jsonObject[0].recordTypeId); 
            }, 
            {objName: component.get('v.objType')}
        ); 
    },
     
    onChange : function(component, event, helper) {
		var value = event.getSource().get("v.text");
        component.set('v.selectedRecordType', value);
	},
    defaultCloseAction : function(component, event, helper) {
        //$A.util.addClass(component, "hideModal");
        $A.util.addClass(component, "slds-hide");
    },
    onconfirm : function(component, event, helper){
        //alert('confirm get called');
        var selectedRecType=component.get('v.selectedRecordType');
        alert('selected recordtype:'+selectedRecType);
    }
})

SkDemoRecordtTypeSelectorHelper.js

({
	callToServer : function(component, method, callback, params) {
        //alert('Calling helper callToServer function');
		var action = component.get(method);
        if(params){
            action.setParams(params);
        }
        //alert(JSON.stringify(params));
        action.setCallback(this, function(response) {
            var state = response.getState();
            if (state === "SUCCESS") {
                //alert('Processed successfully at server');
                callback.call(this,response.getReturnValue());
            }else if(state === "ERROR"){
                alert('Problem with connection. Please try again.');
            }
        });
		$A.enqueueAction(action);
    }
})

SkRecordTypeSelectionController.cls

public with sharing class SkRecordTypeSelectionController {
   
    @AuraEnabled
    public static string findRecordTypes(string objName){
        string returnString='';
        string queryString='Select id, name from RecordType where sobjectType =: objName and IsActive=true';
        List<sobject> recordList= Database.query(queryString);
        List<RecordTypeWrapper> wrapperList=new List<RecordTypeWrapper>();
        for(sobject sb : recordList)  {
            RecordTypeWrapper rw=new RecordTypeWrapper();
            rw.recordTypeLabel=string.valueof(sb.get('name'));
            rw.recordTypeId=string.valueof(sb.get('id'));
            wrapperList.add(rw);
        } 
        returnString= JSON.serialize(wrapperList);
        system.debug('*****'+returnString);
        return returnString;
    }
    public class RecordTypeWrapper{
        public string recordTypeLabel{get;set;}
        public string recordTypeId{get;set;}
    }
}

TestAppForComponents.app

<aura:application extends="force:slds">
    <c:SkDemoRecordtTypeSelector objType="Opportunity"/>
</aura:application>