I don't think you will have a problem getting a taxi (or Uber) on the day. The majority of passengers will have driven to the port, or will be on coaches.
Those using taxis will just be using them for short local runs, so there will be a constant flow of taxis coming and going.
Which day are you disembarking - it us quite unusual to have six ships in port on the same day.
As an aside, I'm not sure which airline you are using, but I assume you know that Flybe have gone out of business.