Can't comment about the situation at LHR but have worked for timematters in Germany and got a lot of decent long haul offers, mostly all across US and East Asia.

True, a lot of the trips get cancelled since the agency puts out the request for a courier as soon as a customer calls. Then after the customer gets the quote, they often enough bail out considering the obscene costs for OBC services.

Anyway, I had the option to do more trips then I had time for. Probably you will need to look for a local OBC agency working out of LHR.