Finding it hard to progress in the Civil Service?

I joined the Civil Service as a 1st class Law grad, as I liked the work/life balance that it gave/the idea of working in the public sector.

After being unable to get on a grad programme or secure a EO/HEO role, I went for an AO role thinking that I can easily progress to at least EO after my probation.

However, I’ve been here for 10 months now with no sight of progression..my job has given me a lot of customer service experience and even leadership experience(covering for TL) but I’ve been unsuccessful. I thought progression would be easier as an internal candidate but it doesn’t seem that way.

Does anyone have any tips/advice?

I don't work in the public sector, but from what I've heard it's a sector that relies on a lot office politics (less about meritocracy). If you want to progress, consider rubbing shoulders with the "right" people.

Having said that, it's not common for people to be promoted to progress in their role within 10 months. Usually, people send 1-2 years (sometimes 5-10, depending on the role and department) in their role before being considered for promotion. It's a slow game, not a quick one.
At a stage in history where organisations are progressively becoming flatter structures, it's very difficult to get promoted to "top dog" since:

I joined the Civil Service as a 1st class Law grad, as I liked the work/life balance that it gave/the idea of working in the public sector.

After being unable to get on a grad programme or secure a EO/HEO role, I went for an AO role thinking that I can easily progress to at least EO after my probation.

However, I’ve been here for 10 months now with no sight of progression..my job has given me a lot of customer service experience and even leadership experience(covering for TL) but I’ve been unsuccessful. I thought progression would be easier as an internal candidate but it doesn’t seem that way.

Does anyone have any tips/advice?

Bit confused. You seem to be equating your degree class to suggest you should be working at a higher grade. You say you weren't selected for a grad programme or HEO/EO role, why do you think you'd magically be ready for promotion straight out of probation?

We don't know what feedback you had when unsuccessful for EO, nor indeed what EO roles you've applied for.

Progression should be a lot easier as an internal as internal candidates tend to understand the competency framework much better and tend to be better structured in answering competency based questions.

Are your examples cogent at EO level? With a 1st in law you should be better equipped than most in written/verbal communication to explain your examples. The main thing what examples do you have when covering for a team leader.

I have been offered a HEO role and an EO role - I rejected both, due to location. That was 10 months ago, since then I have been an AO and thought I could get an alternative EO/HEO role, but have not.

OK, so your original post is inaccurate.
You have been able to secure EO/HEO roles.