In a company, if a manager says "I am budgeting X for this employee because of this business case", is it still possible that this employee will be made redundant later on because there is no work?
is it still possible that this employee will be made redundant later
Yes, of course. The decision to lay off workers isn't bound by the budgeting process.
I'm guessing that you are either fearful of being let go and are looking for "ammunition" for your case against them. Or that it has already happened.
Unfortunately, you won't be able to use the budget to bolster your case. It simply doesn't work that way.
Sure. A business case is just a plan and plans can go wrong. Maybe the client cancels. Prices change. Life happens.
You will need to contact a local lawyer specialized in employment law to check if it's legal to be laid of under these circumstances, but from a business point of view, an employee can become redundant at any time.