It would be a good idea to cycle the new tank. The new sand is going to contain a certain amount of organic matter that is going to decay and cause an ammonia spike. The size of that ammonia spike will depend on how much organic matter is in the sand and how much new sand you put in the tank. Your live rock (plus macro algae and skimmer if you're using those) may be able to handle the ammonia and nitrite from any decaying matter in the new sand, but I would not say that it would definitely be able to handle it without any adverse effects to your tank's inhabitants.
I wouldn't necessarily expect a nitrate spike, unless there's a condiserable amount of detritus built up in / on your live rock. Swishing each piece in a bucket or tub of your old tank water should help remove a good amount of any detritus that's present. If you'd like to be a little more thorough, you can also blast individual pieces of rock with a turkey baster or powerhead with a concentrated outlet. The amount that is there can be surprising.
Add extra nutrients? I think we may be calling different things nutrients... To me, nutrients are things like ammonia, nitrites, nitrates, and phosphates. In that context, you wouldn't want to add extra of those in most cases. If by nutrients, you mean calcium, alkalinity, and magnesium, then yes, you'll find yourself dosing larger amounts of those in the 125g to maintain the same levels you had in the 55. The need to do so may not come so quickly, as concentrations are not going to change as suddenly in the larger volume of water until the amount of coral and/or coralline algae that is consuming those things grows.
If you're not increasing your bio-load much, you may not need to add all that much to your clean up crew. I would add to the clean up crew conservatively until there's a noticable need for more. For example, initially I'd increase their numbers by only 10 - 20%, then add more as needed.