EmailTemplate template = [
SELECT Id, Name, Subject, Body, HtmlValue
FROM EmailTemplate
WHERE Name = 'My Name'];
templateId = template.Id;

Custom_Object__c obj = [
SELECT Name, Custom_Field_1__c, Custom_Field_2__c
FROM Custom_Object__c
WHERE Id = :my.Id];

Map<String, String> data = new Map<String, String>();
for(String key:new List<String> {'Name', 'Custom_Field_1__c', 'Custom_Field_2__c' })
{
    data.put('{!Custom_Object__c.' + key + '}', (String) obj.get(key) );
    System.debug( LoggingLevel.ERROR,'@@@ key : ' + key );     
    System.debug( LoggingLevel.ERROR,'@@@ obj.get(key) : ' + obj.get(key) );   

}
data.put('User.FirstName',UserInfo.getFirstName());
data.put('User.LastName',UserInfo.getLastName());
String body = template.HtmlValue;
System.debug(LoggingLevel.ERROR,body);
System.debug(LoggingLevel.ERROR,'@@@  template.HtmlValue: ' +  template.HtmlValue );
System.debug( LoggingLevel.ERROR,'@@@   template.Subject: ' +   template.Subject );

String subj = template.Subject;
for(String key:data.keySet())
{
    body = body.replace(key, data.get(key));
    subj = subj.replace(key, data.get(key));
}

I had some other idea to do this.

    String subj = template.Subject;
    for ( String key : data.keySet() ) {
        body = body.replace( key, data.get( key ) );
        subj = subj.replace( key, data.get( key ) );
    }

I had some other idea to do this.

    String subj = template.Subject;
    for ( String key : data.keySet() ) {
        body = body.replace( key, data.get( key ) );
        subj = subj.replace( key, data.get( key ) );
    }

Unfortunately it does not for email templates with letter heads.