Just as a heads up DO NOT wait until you get to Heathrow & grab a taxi there, yes there will be plenty, but the cost is likely to as much as the entire cruise - ok that's a slight exageration, but we had a work colleague travel from Heathrow to our offices last month, a distance of only 57 miles, and they wanted £270 for that journey. He had 1 case, 1 laptop bag and himself. We eventually got the driver to accept £200 in cash !!!
Okay onto the serious stuff now. The transfer is just on 100 miles, and will take around 2 hours, so allow yourself plenty of time.
I would suggest that unless someone comes up with an offer you can't refuse, then enter https://www.google.co.uk/search?hl=e...JeOJ6gG23NCiCw
into your web browser & check out the adverts there, some give fixed prices of £115, whilst others let you enter the details on line to get quotes.
Just make sure that you let them know how much luggage & how many people so that they provide a vehicle with sufficient capacity - many taxis have the secure trailers on them now.
Hope this gives you some options & thoughts.