I am surprised that no one else has mentioned that the supposed electrician has used the earth in the flex as the switched live, meaning that there is no earth protection on the flex. The fan doesn't need an earth but (although, I am not an electrician) I was under the impression that the cable needs earth protection.
The lack of ducting, if is a cavity wall will see much of the vented air running in to the void.
Their website only offers a partial address, other sites that they advertise on list the address as - 9 Sutherland Close Barnet United Kingdom EN5 2JL
They mention that they are TrustMark certified, they are indeed. However, the certification only lists them as electricians.
www.trustmark.org.uk
The address however is listed as - 402 Watford Way London Hertfordshire NW4 4XE
Entering the NW4 4XE postcode into the NICEIC website returns an electrician that lives in a different house. Using the EN5 2JL postcode returns no electrician in that postcode. Google lists Amex Construction at 6 Quinta Dr, Barnet EN5 3AA. EN5 3AA returns no results either. The site does however have the following disclaimer
I can find no mention of Amex Construction listed at
Company's House, I am guessing that the owner is self employed and trading as Amex Construction. Is that specified on your invoice or quote?
Did they charge VAT, if yes, their VAT number should be on the invoice. You can check it at
Check a UK VAT registration number is valid. Prove when you checked a number.
www.gov.uk
All of the above may be perfectly innocent, the guy may have moved home and not updated the info elsewhere.
Have you contacted him? In the event that you decide to take him the small claims court, you will need to allow him an opportunity to rectify things. Judges tend to want the accused to be granted the opportunity to put things right before proceedings take place.
Regarding ducting in a through wall, I don't know if it is a legal requirement. It might be a good idea to ask in the electrical sub forum.
Here's hoping that you find a satisfactory resolution.