Not all money brought to the UK from overseas will attract a tax charge. Generally, it is only remittances of income and capital gains that have arisen offshore while you have been resident in the UK which will be taxed when brought to the UK. Distributions from foreign trusts will however generally be taxed when remitted even if made from accumulated income and gains arising before you became UK resident. Otherwise, if you remit funds you held before you came to the UK, or gifts or other capital you received subsequently, these would generally not be taxable.
When funds come to the UK, there are complex and prescriptive rules that determine whether they represent foreign income and gains or capital. It may be necessary forensically to trace through bank accounts and transactions to analyse the source of the funds remitted.
Setting up separate accounts to segregate different types and sources of funds can make things simpler.
You should take advice at the earliest opportunity if you are planning to move to the UK and claim non-dom status. Ideally, this would be in the UK tax year before becoming UK resident.