Super fancy expensive fork oil is not a major priority for a street bike like it would be for something that gets punished off road, but just like anything else, we have the desire to install the best there is. You can ask each owner and probably get a different answer every time about brand and thickness, synthetic and multi-weight.
For simplicity, I'd go off what is recommended in the service manual taking into account recommendations from the spring maker (Progressive in this case). The factory book will have the suggested Triumph replacement grade and procedure for your bike listed. You can obviously change to thinner oil for softer and thicker for harsh based on your ride preference.
I didn't replace the oil when I changed out the springs, I was careful to remove them slowly so I wouldn't lose any oil (it was less than a year old). I changed the springs with the forks, wheel, brakes etc. still on the bike because I wasn't in the mood to remove the forks just to dump out clean oil. The levels checked out fine afterward so I was good to go with the factory stuff.
The Progressive springs with stock oil resulted in a little softer ride at the top with slightly less dive and firmer at the bottom of travel. Changing the oil weight will of course change the damping characteristics either way however your preferences lead you.
I think that when I do change out the fork oil, I will go one grade thinner just to soften it up a tad. Most likely will use a Castrol product, but that's just me.