Just speaking from personal experience, when the chain breaks down, it is usually not because bridging fails to materialise. It is usually because the buyer / seller at some link in the chain decides for another reason not to go through and it (the chain) can then all collapse.
If your buyer's buyer is looking for bridging, then it usually means he in turn has a prospective buyer.
I would be more worried about one buyer or seller in the chain changing his mind / finding a problem with planning or survey etc. than I would be with the chances of bridging being refused.