You need a better definition of utopia; what you have sounds simply environmental.
The Earth would heal fastest if the human race vanished altogether; there have been several science-backed TV specials on exactly that situation. Of course most pets, cattle, pigs, chickens and other ranched animals would die, even if set free. They can't survive without humans feeding them.
If I remember the specials correctly, without any humans at all, the Earth recovers mot of the way in about a thousand years. Then in about 50,000 years there is no trace left, except possibly some bits of stainless steel and other special metal alloys we have developed for boats and such, which are very resistant to natural passive corrosion.
Perfect Utopia is an extreme, and likely unattainable in real life: It would include the extreme of immortality, for example. So I will talk about what is more plausibly achievable, IMO.
For humans compliant with a low environmental impact, utopia will generally eliminate threats of survival or coercion. No starvation, no lack of shelter (or no need of it), no lack of health care, no lack of safety, no lack of transportation. No coercion (of adults, defined below) to labor to survive (especially since some disabled cannot), or do other things against their will (sexual acts, religious attendance or acts, acts of deference or subjugation, service in a military (which would not exist in world-wide utopia) or as a public servant).
And finally, no lack of freedom (which I will define).
"Freedom" in this context is the freedom to engage in the pleasures you wish, as long as they are not at the expense of any other person's pain or misery or other 'expense' without their consent. For your purposes (the OP) that can include any long term expense to the environment or wild animals.
In particular for humans, the subset of sexual freedom would be high on the list, sex is very important to humans (no matter how much it is hidden or untalked about).
But it also includes other pleasures: In making or observing art, entertainment, music, sports, games (intellectual like board games, to contests of physical prowess from darts to foot races or olympic style events).
Also high on the list, in the same tier as sex, may be diet: people enjoy eating, and the freedom to eat what we want without serious medical consequence would be a component of utopia. If you are going to alter minds with a hand-wavium gas, the single greatest boon to mankind might be to have us severely reduce our urge to eat once we exceed a certain level of body fat, (or other biological markers appear, e.g. diets causing damage that will lead to diabetes might make us stop wanting so much sugar), until the levels return to the normal range. (e.g. eat all the chocolate you want, but you will just stop wanting it if the compulsion becomes a health hazard to you).
That is not an exhaustive list by far. Just consider what people enjoy, that can still be done without any significant environmental impact, and design that system. (Everything will have some environmental impact, even a biodegradable house prevents some plants and worms from living under it, and poses a non-natural obstacle to wildlife. So deciding how much environmental impact is acceptable is part of the design here: Currently we tend to think human lives always outweigh any other form of life, but if that is not true, you must pick a number: How many bird lives does it take to outweigh the life of a human infant? And will the majority of humanity agree, or rebel against that assessment?)
I would define an adult medically by brain development; generally as a mentally competent person of at least 22 years of age that is not a sociopath.
If (as I do) we consider all adult humans as equals, then it seems patently unfair for some group of them (including a dictator, a group of 1) to set restrictions for all regarding consensual adult interactions, in sex, games, sports, entertainment or otherwise, if no specific harm to them or to others can be clearly defined.
*(I personally do not consider the moral disapproval of others definitive harm, or their non-scientifically intuited "chances" of future consequences very compelling. In other words, I require 'harm' to be a measurable consequence in the real world unrelated to the objecting party or the consenting people to whom they object.
I believe in 'informed consent' of adults with a reasonable understanding of the known or plausible risks, and thus I believe, as an equal, if I believe informed consent exists (people have been educated as to the risks), then I have no right to tell adults how to expend their health and life, including if they wish to risk their health or life skiing, skydiving, racing, mountain climbing or adventuring, playing at violent sports or engaging in unprotected or potentially harmful sex. I include this as an aside, but it is part of my definition of 'utopia'.)*
Given the culture described above, I would consider Utopia almost anything permanently sustainable within the renewable energy and resource budget of the Earth, at least while the sun holds out and the current conditions prevail for the next three or four billion years.
As for how quickly such a culture could be formed: Really just a generation or two seems plausible to me, if political, religious, economic and moral objections are put aside, and compliance is assured. Perhaps forty or fifty years, and I see no need to kill anybody (enough would die of natural causes in that fifty years to assure recovery of the Earth, and a sustainable population could be achieved within a century).
I am well informed on the current state of renewable energy, and have done professional research into it (I am a research professor). A 100% renewable world culture is easily within our technological reach, the only barrier is it would be about two or three times more expensive than what we do now.
I would also note one other thing: I would not strive for "recyclable" housing, transportation, etc. I would strive instead for permanence that produces no waste. We have many materials, including natural ones like flint and granite, that degrade so slowly they can last for thousands (even millions) of years without any maintenance or environmental impact whatsoever. We also have metals that can withstand passive erosion/corrosion for centuries, or thousands of years. Build your housing out of that, get your energy from the sun (solar thermal driving steam engines offers the least environmental impact, the best energetic 'fuel' is using those engines to compress plain air to a liquid, and using the energy of the expansion to drive engines or machines). Use hard stone to pave the roads and walkways as well, or (again using solar thermal) various forms of fired ceramics or glass (not smooth and slippery glass, obviously), which in thickness is extremely strong, and with some natural (and non petroleum) additives can be nearly shatterproof even under many tons of pressure.