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I'm looking specifically at Hetman, but I'm willing to look at other brands, but which type of oils are best for maintaining a B&S tuba with the newest linkages?

I generally just use an oil like sewing machine oil for the rotor bearings and linkage pivot points.

You'll get several opinions but I am NOT an advocate of dumping anything down the leadpipe or into the tuning slide tubes other than a VERY light oil like lamp oil. Heavier oils tend to migrate into the clearances between the outside of the rotor and the inside of the rotor housing where they will cause the rotors to become sluggish.

I used to use Alisyn, but over the past few years have changed to Yamaha Synthetic Light Valve Oil which I really like. I've used (and use) this on both piston and rotary instruments, for valves and linkages. On my older 1924 horn I use Yamaha Synthetic Vintage Valve Oil.

A lot of this is personal preference after you experiment with a bunch of stuff. I prefer totally odorless/tasteless synthetic oils. Some don't mind a little garnish of kerosene. Some worry about economy. There is some degree of ideology and brand loyalty. Have a good time with your choice.