Personally I think that much of what people are at times calling for in the press is bravado and posturing (the press and various opinion pieces, I don't direct this at anyone here). I think the reality is that escalation against Isis would have support in Britain, but not boots on the ground. I also think that directly going after Assad is something public opinion in Britain just won't support. Bravado and pride aside, I think we are very weary after Iraq and Afghanistan, and the prevailing view seems (to me, I may be wrong) to be that western intervention has allowed current circumstances rather than limited or helped reduce or delay or contain events. The result is that going after Assad is a very very hard sell. Blurring going after Assad and going after Isis seems ill advised from a political standpoint.
I think we are also, as a population, very weary of things like drone strikes and no fly zones. Libya has a lot to do with this I think, along with a weariness of legalities and normalisation of drone strikes. I can't find it now, but a piece in the daily mail recently touched on this in the jihadi John context.
Essentially, public opinion seems to allow for attacking Isis, but not Assad.
I think Iraq attacks ( air exclusively, maybe even limited SF) would be sellable to the public in Britain, but not Syria. The great fear seems to be the absence of a set time frame, unclear objectives (coming back to Assad and Isis again in part), unclear commitment and an horrendous lack of answers as to what the plan would be after that.....or what replaces Assad given the perceived sudden rise of Isis.
Finally, and a very politically incorrect thing for me to repeat perhaps, is that when I speak to people here in London about this, there seems to be a view that the people of the region don't want or 'appreciate' western efforts there. Essentially we would be putting targets on our backs, at our expense, without a likelihood of desirable outcomes..in that view. I can't back that up with polls, but I add that as my perception of popular views here. I may be wrong, but it is the sense I get.