If he is not named on their birth certificates and there is no order saying he has contact or parental responsibility, then you do not need his consent or a court order to obtain visas for them as you are the only person with parental responsibility for them.
However, it is a near impossibility to argue with goverment visa services, you have to comply with whatever they require.
An order in the type that you want should not take 8 weeks - if you are planning to travel imminently you can apply for an urgent specific issues order to state that you are the only person who holds parental responsibility for the children.
You can do so on form C100, which you can find here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/734888/C100_eng_0818.pdf
The cost is £215, unless you are on a low income or benefits, in which case you may be entitled to fee remission, see here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/634329/ex160-eng.pdf
By ticking that you want an urgent hearing, you should be able to get into court within a few days to a week.
Does that assist?