AC has nothing to do with your itinerary, unless you have an AC codeshare on there somewhere that you’re not talking about — and award tickets don’t use codeshares in *A. AC’s baggage allowance will not apply.

MSC doesn’t apply to flights to/from the US, except at the carrier’s discretion, and would be UA anyway. The DOT rule is first marketing carrier, which, again, will be UA unless there is a weird codeshare here. You will get UA’s allowances.

The relevant rules say nothing about mixed cabin allowances, because I maintain that they were written by people who don’t travel. In practice, the answer to your question depends upon where your bags are checked, which — in turn — depends upon the timing of these flights. If you are taking a redeye flight from EWR, you will most likely get your bags tagged to TPE and get the business class allowance. If you are taking a daytime flight to LHR and then spending the night, your bags will be tagged to LHR and you will get the economy allowance.