Toward the end of the first episode of the new season, we see the Doctor travel to a charity shop to buy some clothes. She also seems to have had her ears pierced and her hair and makeup professionally done.
Noting that the real filming location of the shopping sequence is actually in Cardiff, not Sheffield, that same road contains at least two professional salons that offer nails, hairdressing and ear piercing. The closest is Beauty Nails, Cardiff.
As to the (out-of-universe) meaning of the earrings, this was discussed in an interview with the BBC
Q1. What's the story behind the ear jewellery?
A: Jodie and I decided it would be great to have a single earring. I decided it should be bespoke and so needed a jeweller to
help us design and make it. We met in his workroom and the three of us
discussed what the earring should represent and we decided it needed
to start at the top of Jodie’s ear and travel to the ear lobe. The
story is that the stars are where the Doctor comes from, the chain is
the journey and the hands of peace represent her love of humans. I
wanted to make the hands two different colours to represent
inclusivity so we decided they should be silver and gold. Jodie
offered to pierce the top of her ear but we didn’t think that would be
necessary and the jeweller did an amazing job of crafting a piece
which clips over the top of her ear then the hands go through her
pierced ear lobe and the chain is attached to the back butterfly. It
took about 5 weeks from the first conversation to the end result and
we were thrilled with it.
Ray Holman - Costume Designer Q+A