Thank you to all who replied and helped me plan. Here's what I did:
Day 1: Landed in London. Roamed around the gardens around Buckingham palace, saw the change of guards, National gallery, gardens around Westminster, and lot of free-willed city walking.
Day 2: Trip to Warwick castle, Oxford, Avon. Suggested by @travelhorizons in the replies and he/she definitely got a commission out of it. But nonetheless, a good day trip. Explored Notting Hill in the night.
Day 3: Natural History Museum, V&A museum, quaint little bookshops around the museums etc.
Day 4: Tower bridge, Westminster Abbey, Tower of London, restaurants etc.
Day 5: flew to Aberdeen. Explored the city on foot, the 2 museums there, Footdee, Seaton Park
Day 6: Took bus to Ballater (short but beautiful hike there), and then Braemar Castle. Returned to Aberdeen in evening.
Day 7: Train to Edinburgh. Explored the Castle, National Gallery, city on foot etc.
Day 8: More museums and National Museum, city on foot. Flew to London.
Day 9: V&A again (can go there again and again). More quaint little bookshops in the city. Flew out of London in the evening.

Sounds like a great great trip with a wee bit of Scotland after all!