Bruntsfield and Polwarth both have excellent direct bus routes for both EU and Heriot Watt.
The services to EU are by the long established No 23 and 27 Services which will take you direct from either Bruntsfield or Polwarth to Forrest Road at 10 minute intervals.
You can view a timetable and route map for each of these services on the Lothian Buses Website
Lothian Buses offers local bus services 25, X25, 34, 45 and N25 (night bus) to the Edinburgh Campus. Service 25 is the most frequent, leaving the city centre every 15 minutes during the day, and every 30 minutes after 8.00pm. The journey takes about 30–40 minutes from the city centre, depending on what time of day you’re travelling.