Certified GKH Gardening Expert
Depending on the type of lemon, and where it was grown, along with which pollinator they used, you may never see a fruit from your lemons, or it may be bred with a wild type leaving you with an inedible fruit. Sometimes they breed true, sometimes they don't.
However, you should still see flowers anywhere from 6 years to 10 years, depending on its lineage. This article will help you to grow lemon trees properly: https://www.gardeningknowhow.com/edible/fruits/lemons/how-to-grow-a-lemon-tree.htm
Papaya is a different story. You have three types- Male, female, and bisexual. The first two types need both in order to pollinate and fruit. The last option (which most are) is self fertile. You should see flowers at about 6 months or so from seed. They flower relatively fast. This article will help you to care for the tree: https://www.gardeningknowhow.com/edible/fruits/papaya/growing-papaya-fruit.htm