Which are the top universities in UK or US to pursue robotics engineering?

UK PR also called Indefinite Leave to Remain usually takes 5 years if you're on a work visa like the Skilled Worker Visa. However some family visas (like being married to a UK citizen) or long-term residence may allow you to apply sooner, around 2 years.

For getting UK Skilled Worker visa, you  must have a job that pays a minimum of £41,700 per year or standard salary for that job - whichever is higher. However some jobs have low salary requirements like  healthcare workers, PhDs or workers under 26. These people might get qualified with salaries as low as £25,000.

Students applying from outside the UK need to pay the following UK work visa fee -

• For up to 3 years - £769 per person 
• For longer than 3 years- £1,519 per person 

Students earn 60 ECTS credits for one full year of study in Europe. This makes 30 credits per semester. A 3-year bachelor's comes to 180 credits total. A four-year Bachelor's degree equals 240 credits total.

The answer is - contact your supervisor. Some universities also share the process to getting your PhD funded by the university. You must contact your PhD supervisor for the same, in most cases, a PhD scholarship is awarded at the time of accepting your application. If you don't get a scholarship from your university, you can also apply for an external PhD scholarship in UK. The process to apply for each scholarship is to visit their respective websites and follow the steps as indicated.