SF_10.1, SF_10.2, SF_11.0, SF_11.1, SF_11.2, SF_12.0, SF_12.1, SF_12.2, SF_13.0, SF_13.1, SF_13.2, SF_13.3, SF_14.3 - https://docs.sitefinity.com/for-developers-apply-page-templates

ApplyPageTemplates_SetTemplateToPageNativeAPI.cs

using System;
using System.Collections.Generic;
using Telerik.Sitefinity.Modules.Pages;
using Telerik.Sitefinity.Pages.Model;
using Telerik.Sitefinity.Workflow;

namespace SitefinityWebApp
{
    public class ApplyPageTemplates_SetTemplateToPageNativeAPI
    {
        public void SetTemplateToPage(Guid pageNodeId, Guid templateId)
        {
            PageManager manager = PageManager.GetManager();

            var initialPageNode = manager.GetPageNode(pageNodeId);

            PageData pageData = initialPageNode.GetPageData();

            var temp = manager.EditPage(pageData.Id);
            temp.TemplateId = templateId;
            manager.PagesLifecycle.CheckIn(temp);
            manager.SaveChanges();

            var bag = new Dictionary<string, string>();
            bag.Add("ContentType", typeof(PageNode).FullName);
            WorkflowManager.MessageWorkflow(pageNodeId, typeof(PageNode), null, "Publish", false, bag);
        }
    }
}