I went through this a few years ago. In our case, my husband's ex had remarried but not in the temple. He requested a cancellation because she wanted to eventually get sealed to her new husband. This was approved within about 3 months. She did not then have to request permission to be sealed other than getting a living ordinances recommend from her bishop for herself and new husband.
The church offices do prioritize according to planned marriage dates, but I would be reasonable and give them at least a couple of months from the time they receive the paperwork and letters. Paperwork from both sides must be received for them to start. In other words, if both of you are requesting cancellations or clearances, the paperwork timing must be synchronized by stake presidents (after the bishop is done with his part) working together. Obviously, this isn't necessary if only one party is requesting cancellation/clearance.
You must request letters from previous spouses and they must either directly refuse your bishop or not respond within a couple of weeks to his written request. If the ex can't be found, you will include that in your letter and paperwork when submitting your request to the first presidency. This part usually takes several weeks. Appointments are made with both bishops and stake presidents. Also, ex's are rarely compliant in this regard. Mine said he would send a letter, but didn't. My bishop contacted him again and he said he would send it again, but didn't. This made it drag out about 3 weeks before my bishop made a note of it and moved on.
If you are refused the cancellation/clearance, you can reapply in a year. I have several friends who have gone through this process. One was denied and they were married civilly. A year later, they reapplied and the clearance was granted.
Cancellations are not granted to men unless the ex-wife is agreeable and in a position to be sealed to someone else. As stated previously on this thread, the couple is not tied to each other in eternity if they couldn't get along here. Personally, I like to think of me and my ex as living on the same street in eternity as my daughter. None of us will be there anyway unless we can be nice
Most often, men are given clearances to be sealed to another and women are never given clearances, just cancellations. This leaves the ex-husband without a saving ordinance. Women are protected in this matter.
Civil marriages are no longer performed in the temple unless BOTH old spouses are deceased. This policy changed in 2004. Personally, I see no point in having a marriage in the temple unless the ordinances are going along with it...seems kinda like a waste of a sealer's time.
As far as children go, the first presidency will NOT undo that sealing unless the children are legally adopted by the new spouse and/or permission from the ex is given.
My cancellation letter states, "Children born in the covenant or sealed to parents are assured the right and privilege of eternal parentage based upon their individual faithfulness and agency. If such children remain worthy, these birthright blessings remain secure in the event that the sealing of their parents as husband and wife is canceled. For each of us, the realization of eternal blessings is conditioned upon personal worthiness and individual agency. Such blessings, including our eternal family relationships, will be determined by our wise and loving Father after we have completed our mortal probation."
Do not allow concerns over what will happen to the sealing to your children to sway a decision to remarry in the temple. Children will ALWAYS be blessed by your righteous choices and certainly, the decision to marry in the temple is a righteous decision.