You said "you can't have an HSA if you are eligible to contribute to an FSA," but that's not quite accurate. The fact is you can have an HSA, but you can't contribute to it if you have an FSA or any health coverage besides the HDHP. However, this does not invalidate your entire year's worth of HSA contributions; the HSA contribution limits are prorated based on how many months of the year you are HSA-eligible. See this question-and-answer for details on how to prorate the HSA contribution limits for partial-year eligibility. If you find that you have already contributed more to your HSA than you should have, you can request an Excess Contribution reimbursement from your HSA provider.
Even after you lose your eligibility to contribute to your HSA, you can still use the funds in there for qualified out-of-pocket medical expenses.
As for your LPFSA, you should be able to use that through the month of December, as well as your wife's new FSA. However, anytime you use any of these funds, you need to make sure that you aren't double-dipping. You can only be reimbursed by an HSA/FSA for expenses that are not covered by insurance, and you can only be reimbursed for expenses once. If you find, after a medical expense, that you have more money in your pocket than when you started, you are doing it wrong. :)
For the month where you are covered by two health insurance plans, health insurance companies employ something they call Coordination of Benefits (COB), where they work out among themselves who will pay for what expenses. This is so that they don't end up both paying for the same thing and ensures that you don't profit from a medical expense. Essentially, what will happen is that they will decide which plan is the primary coverage and which is secondary, and the primary coverage will be responsible for the expense, with the secondary insurer paying only what the primary doesn't cover. There are complicated rules in place to determine which one will be the primary (see this article on netquote.com for details). If you end up having a medical expense in December, you probably should call up your insurance provider and let them know of the situation, so the two companies can communicate.