With John Egan, Jack O'Connell and Jack Robinson still injured, the absence of the trio will leave the Blades very short defensively at the weekend.
“Both the players will be missing on Saturday so that's a situation we have to deal with and get through,” said Wilder of Basham and Bogle.
“Hopefully those two boys will be in contention for next weekend's game at Leicester and definitely the last game before the international break at Chelsea in the Cup. It's been a difficult period for us but one we have to adjust and adapt to.”
In reality, with the points gap to Premier League safety still bigger than the number of games remaining even after Wednesday's 1-0 win, the Blades have only pride to play for in the league but the FA Cup is a different matter. They will be long-odds outsiders against a Chelsea team revitalised by new manager Thomas Tuchel, but beat Frank Lampard's Blues 3-0 at Bramall Lane in July, and might hope they are distracted by their push to qualify for next season's Champions League, and to go deep in this seasons.
Basham and Bogle were both injured in last month's defeat at Fulham, the former pulling his hamstring and the latter injured in a controversial incident when Alphonse Areola caught him in the shins on the follow-though of a tackle.
Jagielka's dismissal against Aston Villa only added to the problems, but the Blades decided against fighting it.
The 38-year-old was awarded a yellow card for his foul on Anwar El Ghazi, but that was changed to a red when a video assistant referee review concluded he had denied a goalscoring opportunity. With Kean Bryan square of his fellow centre-back, it seemed hard to call Robert Jones's original decision a clear and obvious error, which is supposed to be the criteria for overturning it via VAR.
But those same shades of grey might have made an appeal unsuccessful too and manager Wilder, who has spoken before about the futility of contesting decisions after games, was persuaded against.
“We looked at it through the night and the next morning but we did but we took advice on it. Yet again we had a difference of opinion but we were told on a technicality the three things they're looking for and consider.
“I think I've been consistent on this, it has to be an unbelievably clear and obvious mistake (to overturn a referee's decision) and I didn't think it was but they thought it was so it was pointless from our point of view (to appeal) having taken advice and spoken to a couple of people and we have to accept that decision.”
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