corporate-wage-slave

The Lima flight is now due in at LGW for 15:52, against a scheduled arrival of 14:20 It's currently (09:40 BST) just north of Puerto RIco and east of Bermuda.

The departing services from LGW to EDI today are
08:40
15:45
21:55
That's it, and normal for a Saturday. LGW to GLA is not going to help, two services at 13:55 and 22:10.

1) The reason for the flight delay is that the aircraft G-YMMR was a few minutes late into LGW from it's previous service from Orlando, and further delays on the ramp at both LGW and LIM. Each of which gradually piled in an increasing delay. The underlying root cause is that LGW's 777s are worked extremely hard, with very little leeway for accumulated delays, and not much scope for swapping equipment around to get out of trouble. A newly refitted 777 will be back in service in the next day or two which will help a lot.
2) No earlier EDI service, see above, I guess your sister could slightly improve timings by going to LHR but if she has luggage she may or may not find that convenient.
3) This looks like eligible for 600€ from what I'm seeing but get your sister to write down exactly what the captain said just before departure from Lima. See the main EC261 thread for more details, it's in the Dashboard.
4) BA's general stance is that food is provided in the lounge (and there is a reasonable hot buffet from 18:00 hrs) so they will have discharged that aspect via lounge access.

For the record, here is the dispatch information with delay code ZY in use on both outbound and inbound. This means a delay caused by the previous service operated by the aircraft. The outbound aircraft left LGW at 13:16 instead of 11:55, took 30 minutes to get airborne (about 15 minutes longer than normal), and arrived in Lima 19:15, instead of 18:30. Which means it was 45 minutes late getting to Perú and then accumulated a further hour on the ground, to departure 1 hr 45 minutes late. The first extract is for the flight returning to Gatwick, the second extract the earlier departure.