(Surprise) visiting a relative that will be studying at University of Essex. What's the fastest way from LHR to the area?
There are several main ways to reach Colchester (University of Essex): by coach (National Express), or a combination of train (National Rail) and tube (London Underground). Depending on which you choose, pricing and time matches variably. I'll provide the options for train and coach, respectively.
By train: Go to London Liverpool Street Station (LST) from Heathrow, there are three ways to do so:
TfL Rail from Terminals 2 & 3 (HXX) or Terminal 4 (HAF) to London Paddington (PAD), then change for London Underground Circle / Hammersmith & City Line to Liverpool Street.
Heathrow Express from Terminals 2 & 3 (HXX) or Terminal 5 (HWV) to London Paddington (PAD), then change for London Underground Circle / Hammersmith & City Line to Liverpool Street.
London Underground (Piccadilly line) from Terminals 2 & 3, Terminal 4 or Terminal 5 (identically named, but separate stations from TfL Rail/Heathrow Express stations). Line change is necessary but there are at least four ways I can imagine this can be done, hence omitting the relevant instructions.
Then take the National Rail train from London Liverpool Street (LST) to Colchester (COL). According to the National Rail journey planner, the journey takes approximately two and a half hours and costs £116.
By coach & Uber: Take the National Express directly from your terminal to Colchester (COL). From COL, hail a taxi. This journey takes approximately the same time, minus ~5-10 minutes, and costs ~£38.
(Both options can be reserved fully in advance.)
Sources: tfl.gov.uk, tfl.gov.uk/plan-a-journey, & ojp.nationalrail.co.uk.