What side of Hyde Park are you staying on? For the south side there should be good Tube access via the Piccadilly line from LHR while for the north side of the park the Heathrow Express or Connect is a good, albeit expensive, option.
I've never flown into LGW on the CX flight and have rarely arrived on CX into LHR but I'd probably choose LHR in this circumstance for the arrival lounge and onward transit options that are likely to get you closer to where it sounds like you're going in town. For my money LGW is most useful for people based in south London, east London, or fairly near the north-south rail connection through the center.
ETA:
Originally Posted by
KACommuter
It only makes sense to use the arrivals lounge if you arrive into Heathrow on the later overnight flight. Even then you end up spending around 2 hours in a fairly utilitarian lounge. If you get into LHR on the earlier overnight flight, I would just take a cab straight into the hotel without bothering about the arrivals lounge. It shouldn't really take you much more 30-40 mins to get to anywhere around Hyde Park if you leave Heathrow at ~6:30 am.
By comparison LGW involves a much longer minicab ride, or a train + taxi ride, which involves an additional transfer. Why bother unless you are staying right next to Victoria station?
I would absolutely use the arrivals lounge for either morning arrival. The hotel may well not be ready for check-in on arrival, so the lounge is an opportunity to shower, change clothes, and grab a bite to eat before setting off for the morning.
Regarding LGW, everyone is forgetting that the Gatwick Express into Victoria isn't the only rail option. If OP were staying in parts of the City or the Kings Cross area, I'd probably recommend LGW and then the Thameslink train. But for west London including the Hyde Park area, LHR probably makes more sense.