I’m so sorry that you’ve lost your mum 
My mum died at the start of the month three weeks after testing positive for Covid. She was in her 90s and had had dementia for the past five years, but it’s still a big loss.
My mum wasn’t religious either, but we had a humanist celebrant. It was a socially distanced funeral in the crematorium, with only ten allowed inside (and no one 70+ or in an at risk group, although I don’t know how they police that). The crematorium also gave us the option of leaving the main doors open for anyone who turned up to the car park to pay their respects. More people that we’d been expecting came to stand outside and we later found out that the crematorium had placed small speakers at the doors, so people could hear the service. My understanding of maximum numbers is dependent on the size of the chapel and the undertaker’s own policy - I don’t think it’s a one size fits all thing.
We had a eulogy, one song, a short period of silent reflection and then the committal. The service was 30 mins rather than the usual 45 mins - we were encouraged to have a shorter service rather than required to do so. They also wheeled the coffin in rather than carry it - the undertaker had told us this in advance, which I was glad about as I otherwise might have been a bit taken aback. The celebrant was really good - he’d never known my mum but took time to chat with us a few times and he read our our tribute word for word. Although everyone there knew my mum’s life, it was nice to hear it.
It’s a personal choice, but it wouldn’t have felt right to me not to have had a eulogy. You may also need to bear in mind that people might turn up to wait outside, unless you advise them specifically that you don’t want that. If you’re ok with folk being outside, then it might be quite nice for them to be able to hear the service as well.
I think my advice would be to chat with the undertaker so that you’re aware of whatever restrictions are in place and then to chat as a family re what you want. Don’t feel bad about going back to the undertaker with questions and/or for advice - that’s what they’re there for.
As I’m writing this, I realise I’m being very matter of fact - I don’t mean to be as this is a truly dreadful time for you. I was really dreading my mum’s funeral but I got through it (as we do). The time between now and the funeral will be very testing for you, but it’s just all part of what’s a shitty time.
I hope you’re bearing up as well as you can. My thought are with you.