Can a company run a credit check on me without my permission?

Full disclosure I am not personally familiar with U.K. law and am not even a U.K. resident so this answer should not be taken as authoritative in any way. It is simply provided to perhaps help "point the way". If you believe your credit report has been accessed illegally, I would check with a solicitor or barrister to see if they might recommend legal action.

Is it legal to do a credit check without my permission? And what can be done, should I not be informed prior to the check taking place?

A quick Google search seems to indicate that access to your credit report may be fairly limited. The following is from the experian.co.uk site:

"A company must always get your permission to check your credit report. When you apply for credit you will usually consent to the lender checking your report at various stages in their relationship with you [...] [i]mportantly, if a debt is sold on to a debt collection firm, this consent stays with the debt. This gives the collection firm the right to access your credit report to assess your overall financial situation [...] Now, if there is a dispute about whether you owe the money in question then the firm should look into this for you [...] [i]f, however, you actually dispute any connection with the debt at all – perhaps it’s a case of mistaken identity? – you should make this clear to them and they should take prompt action to rectify the situation. If it isn’t your debt then the debt collection firm should certainly not be checking your credit report. [emphasis added]

Experian.co.uk Meta FAQ states the following regarding "Can anyone search my credit report?":

"No, searches can only be made with your consent. This is usually given at the time you apply to a lender."

Experian.co.uk Meta FAQ - Can I use Experian to do a credit check on another individual?:

"No, you can only order your own credit report."

Experian.co.uk Meta FAQ - Do I have a right to privacy?:

"Under article 8 of the European Convention on Human Rights, you have the right to privacy. However, whenever you apply for credit you will give the lender permission to share information about you with credit reference agencies and others. This will be part of the contract between you and the lender. [emphasis added] However, you can contact lenders directly and ask them not to use information about you for marketing purposes."

That said, you can apparently see who has accessed your credit report (at least with Experian):

"Whenever someone accesses or searches your information on the Experian database , a search record, or 'footprint', is created. These records show which company has accessed/searched your personal information and when, and are kept on record for one year."

Finally, while I'm not sure it applies to credit reports, the following was found in the Enterprise Act 2002 (apparently moved and updated from the Consumer Credit Act of 1974). It seems to echo similar language that consent is needed to access personal (financial?) information.

As far as whether you should be notified, this likely then depends on who you did (or didn't) allow access to this information in the first place. Likewise, it's possible certain state agencies might have a bit more leeway in this area than private companies or individuals. Finally, while I wasn't able to find any good evidence to back this up, given this repetition of needed consent, I would suspect that employers are probably limited as well, but that is 100% pure speculation.