Adjust the amount of alcohol. You can add more or less vodka depending on how strong you want the cocktail.
Be careful with the peppermint schnapps. The peppermint schnapps is very concentrated and a little goes a long way. Make sure to err on the side of under pouring it rather than over-pouring it. You can always add more afterward.
Alternatives to heavy cream: you can use half and half or whole milk as alternatives to heavy cream. It will be thinner than a traditional white Russian. Alternatively, you can also use whole fat coconut milk for a vegan white Russian.