For a long time now, I have been receiving several small Bitcoin transactions
You should read
my topic on Consolidating small inputs. In short: you screwed yourself big time by creating so many
dust inputs in your wallet. With high fees, many of your inputs aren't worth the fee needed to send them. There's only one option: wait for fees to drop, and then consolidate your funds.
from a payment system
They shouldn't be sending transactions with such low fees. But you shouldn't be withdrawing such small amounts on-chain in the first place.
to my trust wallet
That's another red flag. There are better wallets out there. By restoring your wallet on Electrum you've exposed your seed phrase to 2 different wallets now. It's better to create a new Electrum wallet, and move your funds there (
when fees are much lower).
Is it possible to speed this up somehow
No. The only options to speed it up are
very expensive.
can someone realistically tell me how long it will take and what these local transactions are?
Bitcoin on-chain transaction fees are a free market. Nobody can tell you when they'll get lower.
"Local" transactions in Electrum mean the server dropped them, usually because the fee is too low and the server's mempool reached it's maximum. This will vary per server.
Any information or help would be very grateful
Read my first link, learn how Bitcoin transaction fees work, and avoid small (dust) inputs. You're having problems receiving your funds now, you'll have much bigger problems sending your funds later.
you will need to use the CPFP method
CPFP is too expensive for dust inputs.
use ViaBTC free accelerator.
There's no point. ViaBTC is out of spots, and OP has no doubt a long line of unconfirmed parent transactions.
I don’t know if it actually works
Then don't post it! There's only one working free transaction accelerator, and that's ViaBTC. But they're very limited when fees go up.