Singapore to London

Would schools allow for entry into Year 11 or is it a definite no?
It will be on a case by case basis. A girl in my school was made to join Yr10 because they did not want her to join Yr11 late in the year. But again it will depend on the school, and if they think shes ready from her Singaporean education.
We heard about entrance exams for some schools - would that be possible for such a case where she is trying to join year 11?
Again it will depend in the school, private and grammar schools are generally the ones that require entrance exams. But what i think is most likely to happen, Is that they would ask for her previous grades.
what are the options for the 'second language' subject?
Depends on the school, you can check what subjects the school offers classes on, in their website,
Would it be possible to take a new 'second language' as a subject in London schools for year 11?
The way languages work in the UK school system is that they will only offer classes in a few languages (depending on the school) In my school they offered French and Spanish. In my friend's school they also offered German.
However, that is only in terms of classes/lessons/teachers. If the student can already speak the language they will pay for the student to just do the GCSE exams without classes/lessons/teachers to help. For example, my school did not offer turkish lessons, but my friend (a native turkish speaker) was able to do a GCSE in turkish.
What that means is your niece will probably not be able to do a language in Yr11 unless she can speak that language already.
How would the GCSE exam for that subject work i.e. what level would the paper be?
Nearly all language GCSEs have 2 levels, Higher and Foundation. The highest grade you can get on Foundation exams are a grade 5. Which one your niece will take will depend on what the teacher thinks.
is there an option to take a different subject, in lieu of the 'Second Language' (for example, in Singapore, it is compulsory for every student to take a second language. Is that the case in London too?
No, languages are not compulsory in London. Except "English Language" which most school require students to do. Also some school might also want students to study a language but thats on a school by school basis.

Hello, this is my opinion as someone who had a sister that started at year 10 however it’s similar.
I know you are able to join in year 11 for most states school as long as they have spaces. The school you write exams for are grammar schools or special schools which I doubt would take people in at year 11.
As regard to second language that depends on the school she ends up going. Each school offer different language so that will be important when looking into schools she can go to: ensure they offer the second language she’s studying now.
In regards to the level the teacher would test her to determine which level she would be. Most tend to just place you in foundation however you can argue your case if you would prefer higher.
For most immigrant they tend to put you in lower levels and change it when you’ve done few assessments however since it’s year 11 you will have to argue your case. I would say print off higher paper exams and let her start practicing. This would help her get used to the exam format and marking scheme. I moved from Nigeria and even though we had similar syllabus and I knew the answers but the way you answer matters.
Let me know if you need more information.