Yes, I think the badges are a great idea. I don't want to assume someone's pregnant or infirm and cause offence, and it's obvious when someone's need is greater than mine.
I found when pregnant I was regularly offered seats on the Tube and buses in London, I don't remember having to ask to sit down once I was showing. It was pretty good overall (except for the bus driver who refused to let me on when it was 80 degrees and I was 8 months pregnant with shopping and my card wouldn't tap.... him, I would cheerfully string up).
I have noticed others be ignored, though, including people who obviously would need a seat, eg on crutches or very elderly. And I have been known to announce that fact loudly down the carriage (if standing myself and so not able to offer a seat) - the joys of perimenopause and no longer giving any fucks.
Anecdotally, the people most liable to offer a seat seem to be young men, though middle aged women are also considerate. The least likely to offer are well to do looking middle aged men in expensive suits, reading the FT.
OP - you did nothing wrong, you needed the seat as much as she did. You could consider asking others on her behalf in future, perhaps, to save her the embarrassment. As pp says, easier to ask on others' behalf than on your own.