You are in a tough spot my friend.
DHL will try to deliver the passport the very next day (i.e. if you give them the passports on 12th, it will be in London on the 13 and they will attempt delivery).
However, when I sent my passport, from what I could see from the DHL tracking website, the courier kept trying to deliver the package after the embassy closed, and it was delivered on the 3rd working day (I also had weekends in between).
FedEX and DHL are both reliable but sometimes stuff like this happens.
All I can do here is tell you what I would have done in your shoes, i.e. sent LVO an email explaining this situation and traveled to my brothers wedding, returned back to Dubai and went straight to the courier from the airport to send my documents (prepared everything photographs, return envelope, etc, in advance). Again this is just my opinion.
5 working days (Friday is a working day in UK) is a risky, but somewhat achievable target to send a package to London from Dubai.
Regards,