The following (reported) facts stand out for me:
(1) Sandra Rivett (the nanny) didn't normally work on Thursday nights. She'd switched her night off that week to go out with her boyfriend the preceding night (Wednesday). No one who'd been surveilling the Belgrave home and learning the household's routines would have expected Rivett to be present that evening.
(2) The light bulb in the light fixture at / near the foot of the stairs had been removed and placed on a chair nearby. The police confirmed the light fixture worked fine once the bulb was re-installed.
(3) Lady Lucan was attacked at the top of the stairs, and (allegedly) only became aware of Rivett's murder when her attacker (revealed to be Lord Lucan) confessed to killing Rivett. She then feigned(?) willingness to help Lucan cover up the murder, and Lucan feigned(?) collaboration to clean up his wife so they could proceed with the cover-up. IMHO Lucan was rattled and improvising as best he could once his wife recognized him. Lady Lucan escaped (which may have been her own extemporaneous plan), whereas Lucan himself fled leaving evidence behind (which further suggests he was rattled and improvising after a plan had gone all to hell).
(4) Lucan had made multiple appointments for Thursday afternoon, but never showed up for any of them. He'd made an appointment to meet someone at the Clermont Club at 11:00 that night, and he called to check on the reservation circa 8:30 p.m. (only about a half hour before Rivett was killed). He wasn't preparing to flee that evening, because:
Officers also searched 5 Eaton Row, into which Lucan had moved early in 1973, ... and searched his last address at 72a Elizabeth Street. Nothing untoward was found; on the bed, a suit and shirt lay alongside a book on Greek shipping millionaires, and Lucan's wallet, car keys, money, driving licence, handkerchief and spectacles were on a bedside table.
https://en.wikipedia.org/wiki/John_Bingham,_7th_Earl_of_Lucan
The scenario that strikes me as fitting these facts and the dismal marital history goes like this:
The recent downturn in his already-desperate financial and personal affairs (bankrupt and a stalker of his own wife) motivated Lucan to eliminate his wife. He set out to attack her in the home to which he still had access. After checking on his 11:00 Clermont reservation he undertook his plan, having traveled to the home site in a borrowed car (the Ford).
The planned attack was an ambush in the basement, facilitated by removing the light bulb at the bottom of the stairs to obtain a darkened kill zone. He viciously attacked the woman who'd walked into his ambush, only to realize he'd killed the nanny who was normally not in the house on Thursday evenings. This realization broke his plan and completely rattled him.
When his originally intended target came to the top of the stairs he began improvising - attacking her, screwing up by having her recognize his voice, then trying to gain her sympathy and cooperation in covering up what he confessed he'd done. The odds of success (in covering up ... ) diminished even further owing to his interaction with his daughter, whose statement would be read at the inquest.
When he discovered his wife had escaped he'd lost even that slim hope of covering up the murder, so he fled without returning to his own residence, gathering his wallet (ID; passport; etc.) and making his planned 11:00 rendezvous at the Clermont. The letters he wrote before abandoning the Ford and disappearing entirely were further attempts to notify others of necessary facts (e.g., the upcoming auction of some assets) during an exit into oblivion (either exile or death) that circumstances (and his own bungling) had rendered unavoidable to as arrogant and self-absorbed a jerk as he obviously was.