As you might expect, quite a traditional public sector environment, with all that that entails. Those looking for a high-octane, edge-of-your-seat company to work for are best off looking elsewhere. Some staff very stuck in their ways, and some of the more academic types can be quite arrogant and big-headed;
Abysmal IT which as often as not hinders productivity rather than improving it;
Some customer-facing roles (e.g. listed buildings or planning teams) may involve dealing with some very angry, ill-mannered people;
However the main con of HE would have to be the appallingly low wages, due to HE being a government-funded body. Cuts to the organisation since the austerity era kicked in means that wages have scarcely risen in that time. Now, no one goes into the heritage sector to make big bucks, but quite simply the salaries are not enough to live on in London (or probably one of the other more expensive locations such as Cambridge or Bristol). It is very common for HE employees to be dependent on their partner's income, with the other half doing the 'real' job which earns a decent salary. Management are aware of the wage issue but in many ways their hands are tied by government paymasters (culture and heritage always being the lowest priority of course). HE is trying hard to attract people from beyond their traditional white middle-class demographic, however the poor salaries on offer directly conflicts with this aim. It is a real shame as many talented, enthusiastic people from less well-off backgrounds end up leaving for better-paid positions elsewhere;
Allied to the above would be the poor opportunities for professional growth. Staff cuts means there's little on offer, and when vacancies do arise competition can be fierce. The promotion process is not very transparent and favouritism clearly plays a role in some appointments. Some roles can be quite repetitive and offer little variety or opportunity to learn new skills.