If under the effects of a Fly spell, can you move at a speed of 60 feet underwater?
A character's speed is:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.
Using the swimming rules you can use this speed to swim, but it costs 1 foot extra per foot:
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling.
A creature can have various special speeds such as "swim speed", "climb speed", "burrow speed", etc. However, these only apply to that one particular type of movement. Not for climbing and swimming:
You ignore this extra cost if you have a climbing speed and use it to climb or a swimming speed and use it to swim.
Nor for burrowing nor flying:
A monster that has a burrowing speed can use that speed to move through sand, earth, mud, or ice. A monster can't burrow through solid rock unless it has a special trait that allows it to do so.
A monster that has a flying speed can use all or part of its movement to fly
Climb speed is for climbing. Swim speed is for swimming. Burrowing speed is for burrowing. Flying speed is for flying.
There is no indication that these speeds are interchangeable. You cannot use your swimming speed to climb, you cannot use your burrow speed to fly, you cannot use your flying speed to swim.
Swimming is not flying
As far as I am aware, flying is never defined as a game term, which means the word takes its normal English meaning: moving through the air under one's own power. I don't think most people would describe underwater movement as flying (except metaphorically).
You can probably "fly" at half speed underwater
The rules for swimming state that:
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb, or a swimming speed and use it to swim.
It doesn't say you need to use your walking speed, it just says that you need to use twice as much movement as normal. Hence, you could probably use the 60 feet of flight-based movement from the Fly spell to travel 30 feet per round underwater, which is still an improvement over the 15 feet per round that you'd get with the typical walking speed of 30. (But you should ask your DM how this will work in their game if you're making a plan that relies on it.)