Roughly speaking, an hour. It depends on exactly when in the morning because if you're talking 7am then the roads through the centre of town will be clear and you'll be there likely 45-50 minutes. However, if it's after about 9am, and definitely after 10am, the roads will be busier and it will be an hour.. possibly a few minutes more depending on traffic. I've taken a car to LHR through central London many, many times (starting about a mile north-east from where you will be) and it's never taken more than 1h 15m.

As for which firm, getting an Uber would be my recommendation. However, if you want to pre-book then Addison Lee are ultra-reliable, albeit a bit more expensive.

By the way, you have other options but I'm assuming you already know about those, e.g. taking the tube from nearby Holborn which will take you direct to LHR, although the journey is long. Or a car to Paddington station where you can get the Heathrow Express train. For minimum hassle though, getting a car all the way is good.