I would play it by ear - certainly no rush to decide in the first week.
I haven’t worked for a while (it’s complicated - not strictly due to MS, although definitely a factor).
But I was diagnosed whilst still working. I told HR (didn’t have a lot of choice about that one, because my critical illness claim had to go through them), my immediate line manager, and my immediate colleagues - i.e. people who might be impacted by the fact I wasn’t getting through as much work, and who also might know me enough to wonder what was wrong. I thought the truth was better coming from me, as I didn’t want any rumours starting (She’s pregnant, she drinks, she’s got cancer, or whatever.)
I didn’t tell everyone I came into contact with over routine things. I told one customer I had a pretty good relationship with, because she was based quite far away, and I don’t drive. It was getting a bit of an ordeal to visit her on public transport.
I didn’t want her to think that anything had cooled between us, making me not want to visit, so I explained I was ill, and asked to deal by phone or video conference where possible. I didn’t tell another customer I’d never liked too well - I didn’t feel I owed it.
So see how it goes. If (when) you become friendly with people at work, you may find you want to tell them.
One word of warning, though - once the genie is out of the bottle, you can’t put it back in.
So do not think you can tell just one or two people, swear them to secrecy, and it will never go any further.
I think at least one person here has told a colleague “in confidence”, and been hurt when that confidence was broken. But to be fair, that’s a risk you take. I don’t think there was any malice involved. I think the colleague, knowing the person had MS, became concerned when she was off sick, and alerted someone. So it was all from caring motives, but obviously not the way the person had wanted to handle news of their MS. Once the info is out there, you cannot always control who gets hold of it, or what they make of it.
Overall, I still think it’s better to be honest, and nip any rumours in the bud, but be choosy who to tell, and when. I don’t think you need to make a public announcement on Day 1. Once you’ve settled in, see who you feel comfortable about telling.