Unite said the decision was a "bitter blow" to staff and customers and accused the bank of "walking away" from local communities.
Lloyds said 29 Lloyds Bank and 15 Halifax branches would close, adding that customers were carrying out "significantly" fewer transactions at these locations.
The Sedgley branch is in Dudley Street and its closure will leave Barclays as the only remaining bank in the village.
Its Quinton branch is in Hagley Road West and Rugeley's Halifax branch is in Market Street.
Vim Maru, retail director for Lloyds Banking Group, said: "We've also seen our digital banking customers grow by over four million in five years, to almost 18m, of which 13.6m also choose to be active app users.
"This means that, like many businesses on the high street, we must change for a future where branches will be used in a different way, and visited less often.
"We'll continue to invest in our high street presence, as this week we're opening a new concept Bank of Scotland branch in Edinburgh, the only bank to take up residence in the new St James Quarter.
"Importantly, we'll continue to give our customers a choice in how they bank, with branches sitting alongside telephone banking, online and mobile banking, our video appointment services, our cashback through local shops programme, our participation in the industry BankHUB cash initiative and 11,500 post offices, at which our customers can bank and access cash."
Unite national officer Caren Evans said: "The decision by Lloyds to further erode its presence within our communities is baffling.
"The closure of 44 more bank branches will deny our communities essential services such as access to cash and experienced highly trained staff.
"A local ATM is not a suitable alternative to a staffed bank branch.
"In recent times Lloyds has spent significant resource to sell its message of 'Helping Britain Recover'. Unite seriously question how this decision to walk away from local communities promotes this message at a time when the customers will rely on the financial services sector support more than ever.
"Unite does not view the bank branch network as a disposable commodity and the union believes that the branch network has a value far beyond its immediate commerciality. Unite wants to see Lloyds Banking Group invest in the branch network and commit to a meaningful presence in our communities."
Lloyds said of the closures announced on Wednesday that more than a third are branches situated in or around cities and large towns with another branch "very close by".
It added that across the group's branches, transactions had fallen 10 per cent a year in the five years to March 2020, and significantly further in the year since.
Nine out of 10 customers have a branch within five miles of their home, it said.