Let's go back to basics for a moment.
Is this booking already made and is it one ticket per person or two? If it's one then IF you miss the connection the delivering carrier (whoever flew you to Heathrow on the way out) is responsible for finding you an alternative flight to Tampa. Similarly if you miss the flight back to Scotland the airline that flew you from Tampa to Gatwick is responsible for getting you home.
Fine words in theory but in practice getting anything other than the next available seat on the next available flight (which going to Tampa is next day) will require luck (the plane may already be full) and persistence (they can reroute you even on another airline; they don't want to). But you will get there and you wonlt be charged an extra fare.
If, on the other hand you're buying one ticket to get you down to London and a separate one for the flight to Tampa all bets are off. Each ticket is a separate contract. If you fail to check in on time the airline is under no obligation to you at all. You may need to buy a replacement ticket (almost certainly more expensive than the one you had) and if it's the London/Tampa flight you miss you'll need to buy a new return as well.
One a different subject Kings X Thameslink STATION has closed. It has been replaced by St Pancras International. Your route from Gatwick or Luton to Heathrow is therefore made easier by not having to schlep along the streets round Kings X.
HOWEVER. The route between St Pancras and London Bridge is being rebuilt and as a result the link between those 2 points is often severed at weekends. This means Luton to Heathrow is still feasible via St Pancras when the link is closed; Gatwick to Heathrow using Thameslink and changing at Kings X or Farringdon isn't. Check Nationalrail.co.uk for details.