I would do the first one via IAD. IAD is much less likely to have delays than EWR is, and I have taken the UA122 flight six or seven times now and it is always on time if not early. In early December, the flight should be pretty empty, allowing UA to push out the gate early. And unless you're flying on the weekend or Fri, the IAD-LHR flight will likely be empty (20% full or so). YMMV, of course.

Are both the RDU-IAD/EWR flights UA? Or is one (or both) UAX? LHR at that time of day will be incredibly easy to transit through, I don't see any problems there at all for you. Although you will have to walk from the B concourse where UA is to the A concourse for transiting passengers, then back to the B concourse for SN flights. It will take a while, but you should make it in your time.

If you misconnect at IAD, I don't see a problem since you will likely just be re-routed on UA or NH to NRT and then on NH to SIN, getting you into SIN at 11:45pmish the next day. If you misconnect in LHR, you will have to spend the night there unfortunately. But there is a Hilton at T4 that is easy to get to, and usually inexpensive for London, and pleasant for one night stays. But I'd take my chances with 1hr 15 at LHR at 9pm rather than PEK with 1hr for sure.

Good luck.