How long do I have to wait between applying pre-emergent and overseeding?

Leaving enough time between the pre-emergents and overseeding is going to be crucial, since otherwise the new seeds will be affected as well. If you seed first, you would need to wait at least 6 weeks before using a pre-emergent or until the grass is actively growing. If you apply the pre-emergent first, you would typically wait at least 3 months before seeding so that the herbicide won't prevent them from successfully germinating as well (always check label of the product you select to confirm). If there are only certain areas that need overseeding, you could always avoid the pre-emergent for those areas specifically, but it sounds like you have enough weed density where that is not an option. Click here for a quick, helpful video on the topic. 

For cool-season grasses like fescue, it's recommended to overseed in the fall (when soil temps are in the 60s), and then a pre-emergent can be applied in the spring before anual weed seeds germinate.

If you're really eager to overseed but have already applied a pre-emergent, a potential option is to physically disrupt the barrier created by the pre-emergent (e.g. using a cultivator/garden weasel tool). However, this is very labor-intensive and won't keep the pre-emergent active for weeds popping up later in the season.