If you would immigrate in October when the tax year starts in April in the UK, would that mean you would pay less taxes? E.g. a yearly gross salary of £60k would be only £30k income in that tax year, with a personal allowance of £10k.
Yes, in the circumstances you describe, you would end up paying less tax per month in the first year than in future years. The allowance is allocated annually.
However there is also National Insurance which is calculated weekly or monthly, so you wouldn't get any advantage for that.
As a special case, if you are resident in the UK for less than 183 days in a tax year, you might be considered non-resident for tax purposes. That means you can pay zero UK tax. However residence also involves other criteria. You should check up on this, and plan your affairs accordingly. investing in a consulation with a tax accountant might well be worthwhile.
If you do pay tax, then you pay tax only on what you have earned. So you will be taxed the same as someone who earned 30K spread over the whole year, which would be typically slightly less than half the tax paid by someone who had earned 60K over the whole year. However there can be circumstances under which you are also taxed on earnings from before you came to the UK. The country you left might also tax you on your UK earnings after you left (If that country is the US there is no 'might', they will tax you on your worldwide earnings). With most countries 'double taxation' agreements mean you won't pay tax twice on any of your earnings.
For all of these reasons consulting an expert in international taxation as soon as possible is well worth it.
The immigration issue is a red herring: stating the situation in the simplest terms, someone who earns 30k in a tax year will pay less tax than someone who earns 60k in a tax year.