Because poa annua and and fescue are so closely related, there are few products that will kill the poa annua without also harming the fescue. The only one we are currently aware of is called Prograss. Your other options would be to either spot-treat with a Roundup-type product and then reseed with the fescue in those areas, or you can use a preemergent like Prodiamine 65 WDG in the fall to prevent the poa from growing in the first place.
It is recommended to use Prograss EC Herbicide in both a spring and fall application to get the best results. However, you can start with a spring application and still get good results. This product is also labeled for use in rye grasses, so it should work effectively in eagle rye.
For post-emergent control of Poa Annua in cool season grass there's only one that is labeled for it and it's Prograss. You can hit it with a high nitrogen fertilizer to burn it out or wait until temperatures hit 85 degrees and up for it to die out. The best time to treat is in the fall with a pre-emergent, one of the best ones for this is Pendulum or Dimension.
Your other options would be to either spot-treat with a Roundup-type product and then reseed with the cool season grass you have in those areas.
According to the manufacturer and the Prograss EC Herbicide product label, you would not want to reseed Kentucky Bluegrass until 6 weeks after the last application of the product. Prograss should be applied to Tall Fescue in the fall and when overseeding with Tall Fescue turf-type, Prograss application and the seeding may be done simultaneously.
You should not apply Prograss EC Herbicide to Bermudagrass during Spring transition. Per the product label: "The final PROGRASS Herbicide application should be made to Bermudagrass no later than February 1st or 4 weeks prior to breaking of winter dormancy. Late applications (within 4 weeks of breaking dormancy) to turf that will be breaking dormancy may temporarily delay resumption of active growth (spring green-up) of Bermudagrass in the spring." Another popular product for controlling poa annua in Bermudagrass is Certainty Herbicide. Certainty will also work best when the poa is young (before tillering). We do not recommend applying post-emergent herbicides to Bermudagrass during Spring green up. In warm areas, the poa will die out on its own in the summer, and then we would recommend using a pre-emergent herbicide such as Prodiamine 65 WDG in late summer and fall to prevent the poa annua going forward.