This is common, I had the exact same problem with my placement. i accepted my first placement and explained to the boss that I can only commit for 3 months to which he agreed. I also contacted my second placement and informed them that i would like to defer starting for one month so I could spend enough time on my first placement. It works if you play your cards right. Having said that, the jobs you are applying for may not necessarily be as flexible as mine.
In your situation, I would accept both jobs and start the £10k. Nearer to the start of the 2nd, I would inform HR of my intention to leave. It is unlikely the 1st company will ever invite you back in the future but if it's a £10k part-time job, I'm sure they will easily find a replacement.
hamza ahmad luton 786 hertfordshire hamzaahmad786