I would expect that your minicab would be more likely to park in the short term car park. The driver would then meet you at a suitable point in the terminal and then help you take your bags over to the minicab.

Normally my driver meets me at Costa Coffee directly opposite where arriving passengers come out into the arrivals hall after going through customs. You could ask yours to come up the escalator to the BA arrivals lounge (see description above) and meet you at the lounge reception, which would be much easier. Just make sure that the cab company gives you his mobile phone number.