It's a strange thing, but if a vector (e.g. a character vector of labels) includes an object created by expression(), the whole vector appears to be treated as an expression:

# create a simple vector with one expression & one character string
label.vector <- c(expression("exponent"^1),
                  wrap_format(20)("another also long label"))

> sapply(label.vector, class) # the items have different classes when considered separately
[1] "call"      "character"

> class(label.vector) # but together, it's considered an expression
[1] "expression"

... and expressions are always left-aligned. This isn't a ggplot-specific phenomenon; we can observe it in the base plotting functions as well:

# even with default hjust = 0.5 / vjust = 0.5 (i.e. central alignment), an expression is 
# anchored based on the midpoint of its last line, & left-aligned within its text block
ggplot() +
  annotate("point", x = 1:2, y = 1) +
  annotate("text", x = 1, y = 1, 
           label = expression("long string\nwith single line break"))+
  annotate("text", x = 2, y = 1, 
           label = expression("long string\nwith multiple line\nbreaks here")) +
  xlim(c(0.5, 2.5))

# same phenomenon observed in base plot
par(mfrow = c(1, 3))
plot(0, xlab=expression("short string"))
plot(0, xlab=expression("long string\nwith single line break"))
plot(0, xlab=expression("long string\nwith multiple line\nbreaks here"))

Workaround

If we can force each label to be considered on its own, without the effect of other labels in the label vector, the non-expression labels could be aligned like normal character strings. One way to do this is to convert the ggplot object into grob, & replace the single textGrob for y-axis labels with multiple text grobs, one for each label.

Prep work:

# generate plot (leave the labels as default)
p <- ggplot(df, aes(x = levs, y = vals)) + 
  geom_bar(stat = "identity") +  
  coord_flip()
p

# define a list (don't use `c(...)` here) of desired y-axis labels, starting with the
# bottom-most label in your plot & work up from there
desired.labels <- list(expression("exponent"^1),
                       wrap_format(20)("another also long label"))

Grob hacking:

library(grid)
library(magrittr)

# convert to grob object
gp <- ggplotGrob(p)

# locate label grob in the left side y-axis
old.label <- gp$grobs[[grep("axis-l", gp$layout$name)]]$children[["axis"]]$grobs[[1]]$children[[1]]

# define each label as its own text grob, replacing the values with those from
# our list of desired y-axis labels
new.label <- lapply(seq_along(old.label$label),
                    function(i) textGrob(label = desired.labels[[i]],
                                         x = old.label$x[i], y = old.label$y[i],
                                         just = old.label$just, hjust = old.label$hjust,
                                         vjust = old.label$vjust, rot = old.label$rot,
                                         check.overlap = old.label$check.overlap,
                                         gp = old.label$gp))

# remove the old label
gp$grobs[[grep("axis-l", gp$layout$name)]]$children[["axis"]]$grobs[[1]] %<>%
  removeGrob(.$children[[1]]$name)

# add new labels
for(i in seq_along(new.label)) {
  gp$grobs[[grep("axis-l", gp$layout$name)]]$children[["axis"]]$grobs[[1]] %<>%
    addGrob(new.label[[i]])
}

# check result
grid.draw(gp)