Send them a registered letter and tell them that you intend to file a claim in Small Claims Court in 30 days unless you ahve a response.
You have two problems.. first, establishing your damages and second proving liability. You can report the incident to the environmental health and they can find out if others were sick. I think the chances of proving this are remote and you may be better off just avoiding them in future.
You can file the claim at
https://www.moneyclaim.gov.uk/web/mcol/welcome
It is cheap and can be done online. You may also want to contact BAA and ask them if there have been any other reports. I am not sure if the environmental health department at LHR would track this.