Thanks for the replies. Its really useful to know that the check-in counters are open from 4:30am and it is possible (subject to the mercy of check-in agents/storage space) to check in earlier than 4 hours.

We are currently considering to either:
Leave Cardiff at 5am and arrive LHR around 8am to check-in (early wake-up + the long drive and long day ahead as we have a connecting flight from DOH)
or
Get a hotel room near Terminal 4 and leave Cardiff the night before so that we do not have to worry about the traffic in the morning and waking up too early.