Hi,
The SIM card reader is mounted on the systemboard.
To gain access to the SIM reader you’ll need to remove the systemboard.
Here’s a link to the [https://download.lenovo.com/ibmdl/pub/pc/pccbbs/mobiles_pdf/t440_hmm_en_sp40a26000_01.pdf|hardware maintenance manual] for your laptop, taken from this [https://pcsupport.lenovo.com/au/en/products/laptops-and-netbooks/thinkpad-t-series-laptops/thinkpad-t440/20b6/documentation/doc_userguide|webpage] (select p.2 at bottom of list and then scroll down to find the entry)
Scroll to p.86 to view the necessary pre-requisite steps and then the procedure to remove the systemboard.
That was the easy part.
Once you have access to the reader you’ll have to determine what type and configuration of SIM reader was used and then try to find a compatible replacement part. (I cannot find any information, part numbers, brand name etc, as to what type was used)
Once you have the part you’ll need SMD (surface mounted device) soldering tools and skills to effect the repair.
Perhaps it may be easier to take the laptop to an authorized Lenovo service repair centre and ask for a quote to replace the SIM card reader. They may have access to more information regarding the SIM reader etc and also the necessary tools and expertise to perform the repair.
You may be able to lower the cost of the repair by taking just the systemboard, thereby reducing the labour cost by reducing the amount of time that the repairer spends in dis-assembly and re-assembly of the laptop. The downside of this is that the repairer can’t test the repair and ensure that you have a fully working laptop upon completion and perhaps will offer no guarantee.