My wife and I and some of our friends travel to the UK quite often to see relatives. My wife is there right now. Because of the present troubles in the Middle East, including threats to commercial shipping, have the flight paths of Emirates and other airlines changed to avoid these areas?
Quite generally airlines do not like their planes being shot down, so they constantly monitor the situation and adjust their flight paths as necessary.
In some cases it's the country involved which closes its airspace anyway, in other cases the industry make their own evaluation of the risks. Some large areas like Ukraine or Afghanistan are quite obviously avoided by airlines. In other places it may be more fine-grained, like parts of Syria and Iraq, Israel, etc.
The current threats to commercial shipping are from Houthi rebels in Yemen, quite far from the flight paths from Dubai to the UK. I'm not sure they have weapons which could pose a risk to airliners, but flights which would make sense to go that way have been rerouted to avoid Yemen completely, as can be seen by looking at FlightRadar24 for instance. Here are a few examples:
The few planes over Yemen at this time are all flights to/from Yemen itself, operated by Yemenia, their national airline. There are also a few flights to/from Yemen operated by other airlines, I suppose they evaluated the associated risks.