-Instructing an aircraft to turn right heading 340 is HIGHLY unlikely to be interpreted as FL340 and therefore the risk is extremely low....
From personal experience, the risk is
not extremely low. When given an instruction to "Turn left hdg YYY, Climb FLXXX" it is ALL too easy to get the numbers mixed up, especially with a particular SID controlled by a certain sector out of LHR. In this sector, XXX and YYY are often very similar!
What does help enormously are (in order of effectiveness):
1) Giving the instructions in separate transmissions
2) If not possible due congestion, sticking us on a heading ending in a 5.