No, the Nevilles had no claim to the throne and a blatant usurpation with zero "legitimate" claim is outside the mindset of the English nobility of the time. Even Henry Tudor was nominally the leading Lancastrian claimant at the time of -his- usurpation of the throne.
The only way Warwick could take power is to do what he did OTL, be the power behind the throne, and then marry his daughters into the king he made or his heirs (which he did OTL, his daughter Anne was married to both Edward of Westminister on the Lancastrian side and Richard III on the Yorkist side at different points in time, his other daughter was married to George of Clarence). But he needs to remain in firm control of whoever is king.
The only person I can think of who could be a king he could control is Henry VI, kingmakers had a habit of being killed by the kings they made in late medieval England.