Yes you can swap those words if you want to. They have practically the same meaning, but they sometimes sound better in different contexts. "Nearly" has a slight implication of proximity ("near to a location"). "Almost" has a slight implication of counting ("mostly all of the totality"). But in nearly all scenarios - in almost all scenarios - they're fluidly interchangeable. You probably won't make any mistake if you treat them as perfect synonyms.

The Secret Garden - a great book!

They are interchangeable as far as I can tell but their opposites are not: "not nearly" and "not almost". The former means "far from". For example, "Their personalities are not nearly alike" means they are VERY different. The latter is merely the opposite of "almost".

Agreeing with Dan. In some situations, one may be more appropriate than the other but in this situation they are interchangeable/equal. The main difference is that “nearly” is only used in positive forms where “almost” can be used in both positive and negative forms. For instance: “There is almost nothing left.” I’m sure you will find people who say “There is nearly nothing left.” but it’s uncommon (at least in the US) “Almost nobody came to the party.” These are places where you wouldn’t swap them.