I think, as I understand it, that this is a major difference between the US and the UK, Across. If I'm getting my understanding of the US and the importance of the constitution correct, it's simply unthinkable to deploy troops on US soil. (Hence the existence of the national guard). In contrast, the military over here has been used historically for tasks as mundane as strike breaking - deploying military fire engines in the event of a fire-fighters' strike, for instance.
What they have not been used for, typically, is public order.
This speech from then Minister for the Armed Forces, Nick Harvey, is very interesting. He points out that the army is trained to apply lethal force against external threats, not trained for policing, and lays out very carefully the difference between policing by consent and military force, and argues that the latter is not an appropriate response to civil disorder. So this was ConDem government policy as recently as 6 years ago.
Counter-terrorism operations are (or should be) seen as very different from policing civil disorder - so certainly there are reasons to believe that quite a few politicians in the Tory party would strongly resist any attempt to extend temporary measures to use the army for counter terrorism to encompass policing demonstrations as well. However (and it's a big however), we all know that "mission creep" can and does take place. And our existing restrictions on the right to peaceful protest are among the most restrictive of any countries I know of that can reasonably claim to be democratic.
Incidentally, a quick check of wikipedia shows the threat level has been up at critical on two occasions in the last dozen years or so - 2006 and 2007. As far as I can see, the army were not deployed in the aftermath of the 7/7 bombings in 2005. Armed police were used to cover the G8 summit in Gleneagles, but no hint of the use of the army.