Sorry this isn't going to be the answer you wanted..
told I was unfit for work but would not send me home
I am fit to work just need adjustments for one day.
You're unfit for work through injury/illness not resulting from work activities so should really be following whatever the sickness policy is for your company. The work they requested of you sounds reasonable (something like an office or retail worker being asked to take a 10 mile hike over mountainous terrain when that is couldn't be foreseen as part of the job would be an example of "unreasonable" - just walking around another area of the facility isn't I'm afraid) and you refused to do it. You haven't "shown willing" and therefore the onus is on you to declare yourself sick and leave.
I'm assuming here that your goal in asking to be sent home is to be eligible for pay (also assuming that your employer doesn't offer sick pay beyond Statutory Sick Pay), unfortunately this doesn't apply in your situation as it is you that refused duties first. If (for example) you had agreed to do the work assigned and they had said to you "this isn't working, you can hardly walk - go home!" then it would have counted as you having shown willing and they would have had to pay you (unless you are on a zero-hour contract).
If you have been sick for 4 consecutive days or more (including non-working days) then you are eligible for SSP and can claim that for up to 28 weeks (although it is not paid for the first 3 days).
If your company does provide company sick pay (a.k.a. an "occupational scheme") then check your contract for any conditions that need to be followed (such as reporting methods, deadlines and quantity limits) and if you can meet those then then go home, put your feet up and recuperate! That's what it's there for.
failed to make reasonable adjustments for 1 day
They are under no obligation to do so, the legal obligations around making "reasonable adjustments" applies in the case of disability, pregnancy (during and after), and long-term health conditions such as arthritis - not because you're sick and injured.
threatened with disciplinary
Are you really surprised by this? You've behaved insubordinately, made unreasonable demands and tried to make your sickness/injury their problem. I'm surprised it's only at the "threat" stage to be honest!
Are they wrong to act like this?