Pressure spray works well. Teak oil after tried out.

 

Try applying a mixture of 75% detergent and 25% bleach, boosted with a few splashes of TSP to the teak. This mixture should do a good job of cleaning and lightening the teak. Then go ahead and oil or varnish after it's dry.

 

Bringing back teak is almost an art form. There are others here who are likely more experienced than me but a couple of suggestions - High pressure spraying such as with a pressure washer will remove the soft wood making the teak very grainy and lead to more dirt being trapped in the grain. Similarly, hard bristle brushes will do the same thing. What you want to do is not kill the mildew but remove it in such as way so as to not damage the wood.

 

I'd pressure spray to get rid of the majority of the mildew. Bleach isn't necessary or as effective in terms of removing the mildew... it will kill it, but not really remove it. Once you've gotten rid of the bulk of the mildew, you can wash the surfaces down with bleach, which will reduce the amount of bleach required to do the work.

No matter what you do, you're probably going to have to sand all the teak a bit, since it's been sitting neglected for two years... so the minor damage from the pressure spraying should be covered by the sanding you'll be doing prior to re-finishing the teak.