Make sure you obtain title insurance (enhanced policy) on the property. There many nuances in obtaining an ingress/egress and what is acceptable to not only the county for recording, but the title company, as well. The new lender will close, and the title company will issue insurance, subject to the easement being recorded.....meaning, you can proceed provided your title company provides evidence the easement was recorded prior to loan closing. If you are paying cash, then you only need to worry about the title insurance. There is no reason to wait unless you have separate parties handling the easement and the closing. If that is the case, contact the title company and see what their requirements are - most likely it's a copy of the recorded document and a court clerk's receipt for recording.
Note: not meant to be a substitute for legal advice.