Thanks all. Last time I ended up at LHR instead of LCY was due to to weather cancelling flights and it was with LX,. 10pm ish and they did pay for a cab from Heathrow to home (they even called one for me and gave me a voucher to give to the driver!)

The rebooking options BA gave me were tonight to LHR, tomorrow night to LGW or Monday to LCY. So I assume they would have had to pay for a hotel if I'd gone for option b or c, maybe I should have gone for the LCY flight on Monday 😂
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​​​​​​I'm not happy to be on the tube on my own that late at night.So I think the most sensible will be to try for the HEX
A cab from LHR to home will be over £100 and it's not exactly something I can afford right now if BA won't refund all of it

Still in the lounge at NCE, it's going to be a long trek back...