Yes, in theory.
But in practice, you might be better in saving up for a bit longer and upgrading to a complete new CX bike.
At £184 for the frame and fork, you will need to factor in a new headset (£30) and bottom bracket (£30) and someone to fit them for you if you can’t do that yourself. Chainrings and a cassette could be anywhere from £20-£60. You’ll need the cost of new brakes, plus most likely new cables and housing too. Take into account that the chain and derailleurs on your current bike are probably a little worn anyway.
When you consider new tires and tubes for the cross bike, you’re already getting close to the £500 you’d need to buy an entry level cross bike.