We will be arriving at LHR around 7 am. The dilemma is that we will be unable to arrive at our leased apartment until 2 pm. Are there any luggage courier services from Heathrow that could accept our luggage and deliver it to our apartment later in the day? This will allow us to explore the city for a bit without having to worry about dragging our luggage around.

Thanks in advance to all of you experts!

leave it at the train station where you arrive from airport.

I don't know about luggage courier services, but you might try a Left Luggage service at one of the rail stations. Perhaps there is one near or enroute to your apartment. See below:
http://www.left-baggage.co.uk/services.php

Thank you! We will check out the left luggage service.

Hi,
I think Madex is a good option for you. I have used them and they are good. Here is a link to their site:
http://madexlogistics.co.uk/
Hope this will be useful for you
Have a nice vacation