United are looking for a new head coach after being hit by a double whammy from Hearts.
Four days after delivering a legal petition to Tannadice challenging United's promotion to the Scottish Premiership, Hearts approached the Tayside club for permission to speak to Neilson.
Although the request was initially rejected, Neilson's return to Tynecastle was soon confirmed after the 40-year-old made it clear his heart was set on moving.
United now face the prospect of defending themselves in court against Hearts while looking for a new head coach amid uncertainty over which division they will be playing in next season.
Wright quickly emerged as a leading candidate, although the 56-year-old remains the favourite to become Northern Ireland manager.
Under different ownership, United made an approach for Wright in October 2015 after sacking Jackie McNamara but the Northern Irishman signed an improved contract with Saints and stayed in Perth.
Former chairman Stephen Thompson instead appointed Mixu Paaatelainen and the club were relegated at the end of the season before spending four seasons – at least – in the Scottish Championship.
Wright had won the Scottish Cup with Saints in 2014, beating United in the final, and went on to achieve five top-six finishes in seven seasons before stepping down in May.
Gordon Forrest and Lee McCulloch are now taking charge of pre-season training while United vowed to appoint a new head coach "as soon as possible".
Chairman Mark Ogren said: "Although we are understandably disappointed, Dundee United Football Club are bigger than one person and we will continue to work towards our long-term plan with our staff and players and, of course, our loyal supporters.
"Robbie played a huge role in our journey so far, however, we now look towards the future, with a new leader and renewed hope for our fans."