Riley shared an Instagram post of a selfie along with the caption: "Drinking a toast to the next chapter in Countdown's history… I think you're going to like it!"
The move comes amid rumours that Riley and Robinson do not get on, with the former Weakest Link host calling Riley 'noisy'.
Read more: Anne Robinson quits 'Countdown' after a year
Robinson has since admitted that she has left Countdown over a pay dispute with Channel 4.
The host was employed by the broadcaster on a freelance basis, but they wanted her to join the official payroll.
Speaking to The Sun, Robinson said: "I come from a long line of alcoholic, Irish bandits so to suggest I go PAYE is a bit like asking Philip Green or Richard Branson to join the company payroll.
"I wasn't interested in any offer on that basis. Even if it included financial parity. I will, however, miss out on the Channel 4 luncheon vouchers, which is a big regret."
She also spoke to Emma Barnett on BBC Radio 4's Woman's Hour where she remarked: "I marched so that your generation would be really clever at negotiating your salaries and your pay, and you're all absolutely hopeless at it. Why? It's so easy. There's about six steps to getting the money you want."
Robinson also opened up on the rumoured feud between her, Riley and lexicographer Susie Dent: "It was me wanting everyone to be quiet in the studio. I like everyone turning up on time and quiet, quiet, quiet. So we did have to iron that out. But that's it."
Robinson continued: "I think we all came to an agreement that we would have mics off before we went into a record, which gave me time to have a chat with the contestants — that's all — and remind them what their hobbies are."
Despite leaving the show, the 77-year-old Robinson insisted that she would not be retiring from television.
Watch below: Rachel Riley welcomes baby daughter