Do you guys think this is a reasonable thing to do?
No, not if there isn't something else going on that either you aren't aware of or are leaving out of the description.
An example reason that a company may severely clamp down on access to communications like email is if they are in discussions about buying/selling to/merging with another company. Tight control of this information prior to a public announcement may prevent a number of significant problems.
I am employed to be Head of IT responsible for backups, security, sorting out technical issues, among other things
Clearly you are not "Head of IT" and I don't know if that is your actual position or a title you give yourself as part of a limited IT staff (possibly a staff of one). Clearly the director(s) making the decisions is the actual head of IT. If that is how your job description reads, then you need them to rewrite your job description.
I personally would not continue to work at a place where a job description reads like that and I don't have the means to do my job. I would have concerns that I would be the scapegoat if/when issues result. "It says in your job description you are responsible for xxx, so the failure is your fault."
I have had my Office365 Exchange admin rights and backup access removed by the company directors on grounds that I "could access sensitive personnel information"
Clearly with that access you could access sensitive information. That is always a potential concern for hiring IT staff, but for IT staff to do their job they need access. There has to be a level of trust in the IT staff commensurate to their job expectations as well as sufficient protection for the company. For example, I myself have often found that I get the same (or more) training on privacy issues than medical professionals when working in such an environment. In other cases, an NDA and/or background check was required.
However, I am curious about the "I have had...access removed by the company directors" statement. This sounds like you had access and it was taken away. This is not something that would normally be done if you hadn't done something to warrant such a change. If you haven't, refer to a few paragraphs above in my answer starting at "I personally would not continue to work..."
I feel like this means they do not trust me, and that I cannot do my job properly
If the job responsibilities you laid out are accurate and/or something else isn't going on, no they don't and no you can't.
However, perhaps they view your job responsibilities a bit differently and as such do not believe you need the access you describe. If this is the case, then be thankful. If I don't need access to perform my job, I would rather not have it. It is difficult to blame someone for things like leaked personal information if they don't have access to it in the first place.
This is also the reason why they will not have a domain network as it means I would have access to everything and I would be the only one who knows how to sort a problem out if things went wrong.
Ah, um, wow? What can I say to that? So badly reasoned, I am hoping that is your frustrated translation of their stance.
A properly maintained and administered domain can significantly decrease the risk of something going wrong. That includes proper documentation and training of a secondary or backup admin (even if their primary job function is not IT...like the aforementioned director perhaps?).
Especially since that documentation should provide for disaster planning and for a smallish company I would recommend that include contact information for a third party support resource (local IT service company) - for when you are on vacation and your secondary admin is sick/injured/dead.
Instead they have local standalone machines in a workgroup which everyone has their own PC as a local administrator which in my head is a horrible security nightmare as I cannot do any overall management for updates, security, virus checking, application installation, checking for rogue devices on the network
You could have simply ended this sentence after "nightmare" and removed "in my head." Again, it sounds like you have do not have control of IT in any sense, circling back to you are not the "Head of IT" for the company. If you were the "Head of IT" then you would be able to make decisions on how IT operates for the company.
Your network is only as secure as the weakest link and when each individual is responsible for their own workstation's maintenance and security, there is no standard for maintaining a minimum level of security.