In My Neighbor Totoro, the girls plant some seeds they received from Totoro, and he helps them to grow them really fast in the night (See here). This tree really looks like a mushroom cloud from a nuke. I'm aware of the nukes that the US dropped on Japan during 2nd World War, still I don't get the meaning of this hint.
If you check most of the animes that Hayao Miyazaki wrote/produced they all have environmentalism, pacifism, feminism, love and family themes to various degrees.
For example in Nausicaä of the Valley of the Wind is about the aftermath of a global war and find balance with Nature without repeating the mistakes for the past.
Same thing with Laputa: Castle in the Sky.
Past/Current wars are the central point of those 2 movies, with Miyazaki pushing for letting go of the 'old ways' and adopting a 'Live and let live' way of life.
In the case of My Neighbor Totoro he went with a family theme. But that does not mean that there could not be some hidden symbols in the images.
Japanese people have been marked severely in their life/culture by having 2 atomic bombs dropped on them. Many manga/anime are based on post apocalyptical worlds as a result of it. Others just hint at this 'scar' that Japanese people have. Take for example Grave of the Fireflies in which it was only recently discovered that the movie poster actually shows a B29 Bomber plane flying above the children when everyone thought it was just a shape of clouds.
Considering this, it is possible that Miyazaki drew the shape of the tree as a mushroom cloud without even realizing it.
This covers the point of view that the tree was indeed drawn to look like a mushroom cloud.
Now, what you have to consider is that a lot of trees actually do have a mushroom shape (that's how kids mostly draw trees).
Also Japanese people venerate the largest trees. Often building small shrine around or next to them.
In the case of Totoro, Miyazaki actually made that tree stand taller and separate from the others to show it as a kind of 'king' of the trees. Worthy to be protected by the guardian of the spirit Totoro. It grows so tall and strong from the love/innocence of Mei and Satsuki.
Personally, and considering the way Miyazaki approaches war in his work, I believe the second hypothesis (a normal tree without any special war-related meaning) to be the most probable one.
It is definitely a mushroom cloud except that it’s one that brings life instead of death. You see mushroom cloud-like imagery and symbols of radioactive destruction in a lot of post-WWII Japanese art and entertainment. There’s a whole book by Takashi Murakami called Little Boy which discusses this. My guess is that Totoro is about Japan re-finding it’s identity after the trauma of the atomic bombings. It’s similar to how a lot of American sci-fi and action movies had scenes of buildings getting blown up after 9/11.