Same workflow provided above seems to work on your real data after conversion from TimeZones to DateTime:
using TimeZones, Dates, Plots; gr()
# load data:
using JLD
d = load("data.jld")
pressure = d["pressure"]
dtm1 = Dates.datetime2epochms.(DateTime.(d["timestamps"]))
waterdepth = d["waterdepth"]
dtm2 = Dates.datetime2epochms.(DateTime.(d["timestamps2"]))
# process data:
dtmin, dtmax = minimum([dtm1[1];dtm2[1]]), maximum([dtm1[end];dtm2[end]])
dt_ticks = round.(Int64,LinRange(dtmin,dtmax,10))
str_ticks= Dates.format.(Dates.epochms2datetime.(dt_ticks),"dd-HH:MM")
# plot data:
pw = plot(dtm1,pressure, xrotation=60,label="water depth",c=:blue,lw=3,xticks=10)
plot!(xticks=(dt_ticks, str_ticks), ylims=(997,1015))
pp = plot(dtm2,waterdepth, xrotation=60,label="baro pressure",c=:red,lw=3,xticks=10)
plot!(xticks=(dt_ticks, str_ticks), ylims=(1.6,2.2))
plot_merged = plot(pw,pp, layout=(2,1), link=:x) # optionally add: xlims=(dtmin,dtmax)
