I think that this has been posted elsewhere - if it was not you Mr JMJ forgive me. There is nothing wrong with the HEX if you are prepared to pay and arm and a leg for 15 minutes on a train. I begrudge paying that for winding up in a dump like Paddington! One may get this for - even so £22 to get to central London and then have fares on top to pay seems ridiculous. Unlike most here, I do not automatically get in taxis or Ubers or whatever unless the weather is foul and there is a walk of more than 3 minutes to the Underground.

I think that you may be putting the conveyance ahead of the destination. I respect that you time may be limited, and that will have an impact. I'm a cheapskate cow and would go into London to see sights, visit an exhibition and then head back to the airport and eat and drink in the Lounge for nothing. I find London prices nail-bitingly expensive. Indeed if you have arrived in First, you have an Arrivals Lounge where you can get something to set you up. Personally if a restaurant was indispensible to your plans - I'd go somewhere with a view. The Shard comes to mind. Just my ideas.