If someone else claimed your child inappropriately, and if they file first, your return will be rejected if e-filed. You would then need to file a return on paper, claiming the child as appropriate. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your child was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.
Winner gets the tax benefits; loser gets to pay the IRS back with penalties and interest. The custodial parent almost always wins. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree.
https://www.thebalance.com/claiming-same-dependent-audit-risk-3193030 (audit discussion)
At one point you refer to your partner as husband and another where he is fiancé. If you are claiming the child, the above reply applies. There are special rules for when a non related person (e.g. parent's fiancé) claims a child; most importantly they are not allowed the Child tax credit or Earned income credit. They only get the $500 "Credit for 'other' dependents".
