If your utility company decides it just needs a new meter cover, which is very likely, there is a good chance they will replace for you. (For a fee) if that is the case you may lose power for 15-30 minutes or you may not lose power at all. If they determine the whole meter needs to be replaced, which I find unlikely, they will most likely temporarily replace the cover, and an electrician will have to pull a permit, and schedule with your power company to have power turned off while it is replaced. Normally they can schedule same day shut off and inspection and turning the power back on. It is incredibly important to secure the meter and remove the birds nest, however, I do not believe this is worst case scenario as far as replacing the meter is concerned. First things first would be to have your utility company come look at it and give you their expert opinion.