It is very common for judges to use a case management or allocation hearing to include a stay for mediation. The order might read something like this:
"It is ordered that:
1. The claim is allocated to the [Fast Track / Multi Track]
2. The matter be stayed from [Date of this order] to [Some date in a couple of months time] for the parties to consider mediation or alternative dispute resolution ("ADR")
3. Any party refusing another party's offer to engage in ADR must, within 28 days of the offer to engage in ADR, file in a sealed envelope and serve a witness statement setting out the reason for that refusal. Unless the parties otherwise agree, the statement will be considered to be without prejudice save as to costs, and the sealed envelope shall be marked "not to be opened by the judge until the consideration of costs"
Or, more simply:
"At all stages the parties must consider settling this litigation by any means of Alternative Dispute Resolution (including Mediation); any party not engaging in such means proposed by another must serve a witness statement giving reasons within 21 days of that proposal; such witness statement must not be shown to the trial judge until questions of costs arise."
The references to costs, incidentally, relate to the fact that the judge can impose cost sanctions on a party who has unreasonably refused to engage in mediation. That is to say: one party can be required to pay a higher proportion of the other side's costs, then would otherwise be the case, if they are seen not to have engaged in reasonable efforts to settle the dispute out of court.
If you are a litigant-in-person (i.e. not using a lawyer to represent you), what do you do now?
You need to find a mediator who is available to help you. I am one, and there are plenty more around to choose from. All the court is asking you to do is at least consider mediation. You can wait and see if the other side suggest something, but you might want to seize the initiative and suggest mediation to them. The onus is then on the other side to either accept your offer to mediate, or explain to the court why they refused.
Don't get too hung up on who to appoint, and whether your chosen mediator will be acceptable to the other side. If you find a mediator who you think is suitable, suggest them to the other side and see what they say. It is hard to second-guess this. Either they'll accept your suggestion, or make a suggestion of their own.
I can give you lots of reasons why you should choose me, but really there are very many mediators who could be suitable. Many solicitors, when making a suggestion, will offer their opponents a choice of three mediators, making it hard for the other side to refuse all three. You could do the same.