What is the difference between I am doing lunch and I am having lunch? Both indicate progress of action.
All right, here's one difference: You can have lunch by yourself or with others, but you generally don't speak of doing lunch unless you are doing it with others. "Doing lunch" is slang for meeting someone and having lunch together.
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.