What is the best way to transport a large bike pack and two suitcases from Heathrow to Pinner?

I'm going to assume the bike is not a folding bike, otherwise you would have specified it as such. Given that, you can't catch London buses or the Underground from Heathrow as the former does not take non-folding bikes and the latter not from that station. Here are your options:

• To my mind the best option would be to find or make a friend in the UK with a decent size car. They can pick you up from Heathrow and drop you off with little to no stress.
• Renting a car if you've got other plans beyond Pinner or intend to rent a car anyway is worth bringing forward - less cost than the non-self-powered options and less time required as well.
• Taxis will allegedly generally take bikes but I would suggest booking one and specifying the nature of your luggage to confirm that it will get taken to avoid disappointment. Most convenient of the remainder, but also the most expensive.
• Public transport - Heathrow Express will take bicycles outside of peak hours (7:30 to 10:00 and 16:30 to 19:00), which gets you to Paddington Station. Once there you can get from Paddington to Baker Street and then on to Pinner while bringing the bike, outside of the 7:30 to 9:30 and 16:00 to 19:00 peak hours. This is probably a compromise option. You may need to buy a separate ticket for the bike box.
• Ride the bike from Heathrow to Pinner. Person #2 takes the luggage. Probably the cheapest of the options. Potentially the least convenient, especially if neither of you are familiar with London.

I didn't bother checking private buses because IMO the train transport would trump it, given the shorter changes required (National Express, for example, drops you off near Victoria Station instead of near Baker Street).