Process for getting deposit back - should I be doing something more or waiting?

We're in the middle of buying a flat (finally we've exchanged and have a completion date!) and are staying in short-term accommodation in the meantime.

Previously we rented a house for nearly 2 years but moved out in mid-October to give ourselves more flexibility for moving and pay cheaper rent for a couple of months. I just don't know what the next step is in terms of getting our deposit back. No one has talked about withholding it yet but everything has gone rather silent and I wonder if there's something I'm meant to be doing.

We found the property through a letting agency but the LLs used a separate managing agent (a friend of theirs) to manage the tenancy throughout, which has always just seemed to add an extra layer of bureaucracy. The original letting agency registered our deposit in one of the government schemes. In terms of getting our deposit back (I'm expecting a few deductions but nothing major) I just don't know what stage we're at.

My husband attended the checkout 2 days after we moved out. The write-up noted lots of cleaning was needed for dust etc.. and a few slight marks on paintwork. As it was written into our contract that money for professional cleaning would be deducted from our deposit after we left, and we had already agreed a figure for this with the managing agent, this was unsurprising. I'm certainly not scrubbing any floors if I'm also obliged to pay for someone else to do it. (It may seem odd to do the checkout before the professional cleaning but there had been a dispute with previous tenants who claimed that damage to the laminate floor was done by the cleaners and not them.) Against all the notes about dust, marks etc.. the words "at cost" are written in the right hand column but no totalling has been done.

The inventory company sent copies of the checkout write-up to me, the original letting agent and the separate managing agent. I expected the managing agent to send me an updated note with suggested deductions once the cleaners had been and ask me to agree these. I thought I would then sign this off and formally request the release of the deposit from the letting agent.

However, there's been no contact from the managing agent whatsoever since this time. When I spoke to the LLs about a separate issue (an online shopping order which went to the old address...) they said that she had not yet actually copied them the check-out report.

Should I be chasing the managing agent or should I just put in a request with the letting agent to release my money and let them chase her?

I know from our old neighbours that there was a major set-to with the LL and previous tenants over their deposit so I'm slightly wary of what happens next.

I've got quite enough to do with the purchase, but the deposit money will come in handy soon enough.

Thanks