Good question. The ANobAG status applies when the employer is in a country which Switzerland does not have a relevant social security agreement with. Up until now, that has meant countries outside of EFTA or the EU.
But now we have the UK, which is part of neither, but AFAIK has extended its existing social security agreement with Switzerland. It seems to be a grey area though, with the extension applying to contracts which existed before 2021, and not much clarity on what applies to new contracts/arrangements.
My hunch would be that the old rules which applied pre-Brexit still apply. That is, you cannot get the ANobAG status because the UK and Switzerland have a relevant social security agreement, so your UK employer would have to pay employer contributions to your Swiss pensions, etc. But I can't find anything clear on whether this applies to new employment taken on post-Brexit.
Sorry I can't give a clear answer. On the bright side, your employer may still have to pay their part of your Swiss contributions, which is generally a much better arrangement than you having pay it all yourself. If that is the case (that the old arrangement still applies), then it is your UK employer's responsibility to sort out the admin with Swiss social security. They can appoint you to manage this on company time, or they can use a payroll company, but that's their worry, not yours.
If, on the other hand, the old arrangement no longer applies, then you should be able to register as ANobAG.
I'm assuming the company you are using (and paying 200 francs a month for) is a Swiss payroll company. This is a practical way to go about it, but it should really be your employer paying that 200 francs per month and not you (unless you agree to the arrangement). Have you asked your employer about this? Also, has your UK employer being paying their portion of your pension contributions, accident insurance premiums, etc. so far? Or has the payroll company been docking this out of your salary?
You are paying the 200 francs per month for the privilege of being employed by a Swiss company (the payroll company) instead of a UK company. This means you can claim Swiss employee rights like sick leave, etc. and have all your Swiss social security, pension, and insurances taken care of by someone else.
As ANobAG (if that status were to apply) it would be more complicated. Your only option for a pension fund would be the Substitute Occupational Benefits Institution, and you would have to pay the full contributions yourself. You would have to get your own accident insurance. You also would have to deal with the social security office directly and pay the full contributions yourself.
Let's just say it would be good for you if the old arrangement still applies, and your UK employer is responsible to meet Swiss social security obligations and to pay their portion of contributions on top of your salary.
Whichever case applies, your employer should provide you with an annual salary statement. You can, for example, send them a template for a Swiss annual salary statement and have them fill it out and sign it.