In order to not give the toxic press and populists any more wind in their sales, there is a very fine line to tread now.
None of the other mainstream parties in Europe want to become the slaves to their own local versions of UKIP in the same way that the Conservatives have. That's what they're going to want to be keeping in check.
But you need to distinguish between UK opinion/press, and continental opinion/press. From now on, it's only the latter that matters to the EU, Britain's far right is Britain's problem only. And there is very little sympathy for the UK position right now on the continent - the mood is very much "bye bye, more fool you".
If the EU drives a hard bargain, no one is going to feel sorry for the UK. Everyone understands that EU countries have to protect their interests, and the "evil EU monolith" narrative doesn't hold nearly as much sway as it does in the UK either.
the only thing weighing in the UK's favour right now is the sheer weight of economics. It would be better for the EU to keep the UK in as part of the EEA, that would clearly be the least disruptive option economically.