I have a bit more time now for a proper reply to your question, and I would feel guilty if you had a heart attack.
We don't know (and don't want to know) the exact details of your past financial situation. The types of income, the amount of income, the types of investments you may have, all build a very different picture for each individual.
Someone with a salary below the FEIE limit, and with minimal income from interest in a bank account will have a very different scenario from someone with a salary over the FEIE limit and with PFIC's, trust reporting, etc.
It's up to the preparer you select, the time frame you request, and their set hourly fee schedule as to the cost of coming into compliance. It's beyond the ability of this forum to offer an answer as to "how much it will cost". With minimum income and other sources, it may be as low as £3,000. With complex investments and a higher salary, it may well be above £10,000. Only you and your preparer will know the answer to that question.
There are a number of firms, both large and small, in London. Their rates will vary somewhat. You'll have to do some googling and some shopping around. BUT, do look for those that are dual US/UK qualified preparers.
Once you are established in the position with your UK employer, tax relations with HMRC will happen more or less automatically. Your employer will be able to explain more as regards your exact situation.