I did something similar earlier this year, I think. I was silver and hadn't earned renewal before my membership year end of April 8th. I flew my 4 segments and earned over 600 points between April 8th and May 8th, while I was still technically silver (card expiry was end of May). I received a bronze downgrade pack in May but no recognition of regaining/retaining silver.

I was pretty nervous about it as my partner, who'd been blue and had flown the same flights, received an email congratulating him on reaching silver - but I'd heard nothing. Phone calls and twitter messages to BA achieved nothing to reassure me that I would be silver until 2020, and the website and app still showed my expiry date as 2018. I eventually spoke to someone who assured me the new status would only be updated on June 1st. He thought I'd then be silver until end May 2019 but checked with a supervisor to confirm it would be 2020.

I was watching the website & app like a hawk on June 1st. I dropped to bronze for a few hours, but by midday on June 1st I was shown as silver with a card expiry date of May 31st 2020.

I've not received a proper silver pack but I have received a 'replacement' card with the correct date.

I don't know why my status change had to wait until after my previous card expired, or why I had to drop to bronze for a few hours before I could be silver again. I can't be certain but I think I permanently lost theoretical seating privileges on my bookings during that brief period, despite technically being silver the whole time.

Anyway, I hope your situation resolves automatically, but if not I'd advise you to keep ringing until you get someone who really seems to understand your issue, because the first few people I spoke to didn't have much of a clue.

Edited to add, I was assured that the system would have me marked as silver for lounge access, even if I was only showing as bronze on the app and only had a bronze card. But I didn't feel too confident about that.