What are some of the methods you would employ to improve team work in an environment where there is a lack of trust and cooperation?

Communication Communication, the more you talk, more you have insight from them. What I found that managers loves to sit in their fancy cubicles and hardly find time to interact with and out-side their team. Ask them to speak and share their problems and issues and try to solve them at your best and if not then don't present any rosy picture be sincere to them and highlight the policy matter.

Human beings are very sensitive ! Start the day by greeting each other.
Let the team sit together over a cup of tea/coffee and chat for10-15 minutes.
Respecting each other and appreciating even small job done builds up the team.
Team leader to initiate activities for team members to help each other.
Avoid any team memeber becoming the topic of discussion for others.
Anything related to any individual must be approached directly and not indirectly.
Individuals job related strengths must be appreciated and weaknesses to be viewed for his/her development and not an excuse for demotivating.
Involve the team in major decision making and encourage them to be participative and appreciate their ideas.
Initiate weekly and monthly meetings or as required and involve the team on team performance and together how the task can be achieved.
Encourage them to be very participative in such meetings and they will understand the importance of working together as a team.
If possible have a monthly informal get together for the team outside the office.

At the beginning it must create a passion for the team has to work through motivating and training in communication skills .
And develop a vision and one goal for all of them to achieve it.
And find them respect , and to clarify the meaning of teamwork to them and how to apply it

Why don't you hire me as a consultant to help fixing the problem? I need a pilot project for Employee Engagement or Wellbeing at Work.
That would be a perfect case.

firstly u should plant loyality between the employees & charge the season of competition between them & the results should be profitable , then u meet the employees together to wear experiences from each others & increase their own inventions to produce which is weird & profitable to the company .