Once you forwarded all your paperwork to your solicitor, they could then start work on the drafting the contracts.
They would review the documents, i.e. your ID and check they have some monies on account from you. They would obtain title register documents from Land Registry, copy all the paperwork you sent them, draft the contract and once everything is collated, either post or email everything to the buyer's solicitor.
The buyer's solicitor will review all the documents your solicitor has sent them. They have to "approve" the contract and once they have reviewed all the documentation provided, they start raising enquiries and will draft the transfer for approval by your solicitor. They will also apply for searches (which takes several weeks to receive and may also result in enquiries being made). Reviewing legal documents, all the protocol forms (fixtures & fittings, PIF etc), searches etc does take some time and enquiries will relate to the information revealed in each document.
Ideally, the solicitors would look at the file the minute it lands on their desks, but the reality is that there are lots of other clients' transactions also needing attention.
To be honest, while it would be nice if your solicitor could update you, I highly doubt there has been any significant progress at the moment. This is exactly the time when things go quiet, while enquiries are being raised and dealt with.
You can ask for a weekly update but any more than that and it becomes akin to harrassment, when the solicitor is just trying to get on with the job in hand, during an extremely busy time with Christmas coming up, a stamp duty holiday which has created an influx of sales and purchases, not forgetting the pandemic which has hit the industry, with many staff still working from home or furloughed.
Patience is your friend.