Nope.
Within *A (and I'm assuming flying Y with *G status) the next departure defines lounge access. You're not departing on a *A flight no no lounge.
Also, no *A lounges in the BA terminals so that also works against you. Assuming
T5 there is a buy-in lounge if you really want one. Looks like showers cost extra and no toilets inside from that webpage.
I'm guessing 10:30a arrival and 1:55p departure from a quick scan of schedules. You'll spend the first 40-60 minutes of that switching terminals. After that either the paid lounge or grab a snack in T5 and chill. They'll start boarding about an hour out so it isn't that much time in the terminal once all is said and done.