Uxbridge is close to Heathrow, which means you have genuine choices. Here is a fair look at each, including when we are not the obvious answer.
Public transport
There is no direct rail link from Uxbridge to Heathrow. The usual route involves a bus or a change onto the Piccadilly line further along, which can work well if you are travelling light, in the middle of the day, and not in a hurry. It is the cheapest option by some distance.
Driving and parking
Heathrow parking is straightforward but adds up quickly over a week or two, and you will still need to get from the car park to the terminal. It makes most sense for very short trips.
Drop-off by a friend or family member
Perfectly sensible for a daytime flight. Bear in mind Heathrow charges a fee for using the terminal drop-off zones, so factor that in.
Pre-booked private hire
This is what we do, so treat this as an interested view. It tends to make the most sense when you are travelling at an awkward hour, carrying luggage, travelling as a group or family, or when arriving on time matters enough that you would rather not manage connections. The price is fixed at booking, so it does not change with traffic.
Timing
Whichever you choose, allow more time than the raw journey time suggests. The A40 and the roads around the airport can be slow at peak times. We have a fuller guide on what time to leave Uxbridge for a Heathrow flight.