So, bad news is I've sent a new transaction that will be confirmed in 5 blocks for the same 1 sat/byte price... I assume this will get processed waaaayy later....
You can check the current mempool status (amount of transactions with X sat/B) here:
https://jochen-hoenicke.de/queue/#1,24h.
1 sat/B is
probably not enough for the TX to be confirmed within 5 blocks.
As you can see on the site above, the amount of TX (and therefore also the needed fee rate to get confirmed within X blocks) has gone up a few blocks ago.
This may clean up soon. I do think your TX will still confirm today, but maybe not within the next 5 blocks.
But since this isn't urgent, i guess you are fine.
If you somehow still need to bump the TX, you might also use CPFP to do so.