How common are emergency landings for airliners? Is there really more than one per day at Heathrow?
No, there isn't more than one emergency landing at Heathrow per day

The UK Air Accident Investigation Board (AAIB) defines serious (reportable) incidents in Appendix A of this document.  If we assume that an 'emergency' landing is one caused by one of the incidents in the list, then we can examine the AAIB reports to determine the number of emergency landings at Heathrow (the site supports filters).

I could only find three incidents reported at Heathrow in 2015:


27 July 2015 Airbus A320-232, G-EUYE - In flight fumes
5 February 2015 Airbus A330-323, N273AY - Collided with airbridge 
6 March 2015 Boeing 777-236, G-RAES - Air turnback due to flight crew feeling unwell


Only two of these involved an 'emergency' landing.  2015 may not be a typical year, but a cursory check of the reports for 2014 and 2016 to-date suggests that there are very few incidents.

If you want to get a feel for the number of airline incidents and accidents each day, take a look at AVHerald (you might want to make a contribution to Simon's running costs).  That site reports 29 landings at Heathrow after accidents or incidents, of varying seriousness.  You'll see that, of the thousands of flights each day, there aren't that many problems reported.