Since this page went up, I've always wondered what exactly they mean by the line that I have highlighted in green.

BA World Traveller doesn't have a "Saver" fare - so are they just referring to the EI fare families? If so, it might have been better to write:
Fare Types - All Plus, Advantage, Aer Credit Card, Aer Space (not included in Saver fares) and Economy fares purchased via British Airways

BA does have a "Basic" fare family which includes 2 pieces of carry-on and no checked luggage. I'm wondering whether a BA "Basic" ticket would therefore equate to a "Saver" ticket which does not allow carry-on on board; EI recently changed their handluggage rules and only the indicated fare classes (and status holders) can bring a carry-on on board, with all others being required to check it into the hold (for free) at the check-in desk; at the gate, no bags can be checked free of charge.

In any case, the OP is connecting from/to an interline partner (BA, on a single ticket), so one piece of carry-on is allowed on the connecting EI flights (you must observe the EI limits regarding size and weight, which are less generous than BA's corresponding limits (BA allows up to 2* 23kg in the carry-on bag), or you may be charged extra. That said, it's perhaps unlikely they would ask to weigh your carry-on unless it looks very big or looks like you're struggling with the weight)

I'd be very interested, though, if you bring 2 carry-ons per person (as allowed on the BA legs) whether EI would observe that; however, I think with 2 bags per person you are definitely more likely to have the dimensions and weights checked.

Please do return and let us know how the baggage situation was handled.

And have a good trip and I'm sorry for your loss.