What is the difference between I am doing lunch and I am having lunch? Both indicate progress of action.
Here's another difference which might plausibly be UK-specific: doing lunch can mean making lunch. A family may ask themselves, "Whose turn is it to do lunch today?"
A statement "I'm doing lunch" can mean "I'm in the process of making lunch"; whereas "I'm having lunch" would mean "I'm in the process of eating lunch".
This is in addition to the meaning Robusto has alluded to: someone may make a suggestion "Let's do lunch," which usually means "Let's meet for lunch". However even that could mean (for example, on Christmas morning, when a Christmas meal might take several hours to prepare) "Let's start to make lunch".
Unfortunately there is not enough context in the question to determine the meaning required.