You're going to have to either get a washer that accepts 240V, or change the circuit.

New Washer

Getting a new washer will depend on the manufacturer, dealer, and other factors that are off topic here.

Change the circuit

Modifying the circuit will require knowledge of electrical safety practices, and general electrical wiring principles/standards/codes. If you lack this knowledge, or are not comfortable working with electricity, please contact a local licensed Electrician.

If you decide to tackle this project yourself, here is an overview of what needs to be done. You'll be going from a configuration with two ungrounded (hot) conductors, and one grounding conductor (NEMA 6-20). To a configuration with one ungrounded (hot) conductor, one grounded (neutral) conductor, and one grounding conductor (NEMA 5-20). To do this you'll have to swap out the double pole breaker for a single pole breaker, and repurpose one of the conductors.

At the Panel

1. Before you begin, you'll need to purchase a 20A single pole circuit breaker that is compatible with your service panel, and a blank panel cover insert.
2. Start by turning the power off by flipping the main breaker to the OFF position, then verifying that power is off using a non-contact voltage tester.
3. Remove the service panel cover (be careful working in the open panel, the main lugs are still energized even when the main breaker is in the OFF position).
4. Locate the breaker for the washer.
5. Remove the wires connected to the breaker (should be either black & red, or black & white).
6. Remove the breaker from the panel.
7. Install the new single pole breaker in one of the slots where the old double pole breaker was.
8. Install the blank panel cover insert in the panel cover, so that is covers the empty space left.
9. Connect the black wire that you removed from the breaker in step 5, to the new breaker.
10. If the other wire from step 5 is white, connect it to the grounded (neutral) bus bar. If the wire is red, mark the wire with a bit of white tape, or a white marker, then attach it to the bus bar.
11. Reinstall the panel cover.

At the receptacle

1. Before you begin, you'll have to purchase a 125V 20A receptacle.
2. Uninstall the old receptacle.
3. Connect the black wire to the brass screw terminal on the new receptacle.
4. If the other wire is white, attach it to the silver screw terminal on the new receptacle. If the other wire is red, mark it with white tape or a white marker, and attach it to the silver screw terminal on the new receptacle.
5. Attach the bare or green grounding wire to the green grounding screw on the new receptacle.
6. Mount the receptacle in the box, and install the cover plate.

Once everything is finished, you can flip the main breaker, and the new breaker to the ON position.

WARNING:
This answer is only applicable in the US, and possibly Canada.

Normally, you can never split a 40A supply to serve two loads rated for a 20A circuit. Except here.

This is an allowed exception for supplying oven/range loads. It's in NEC 220.55, referring to Table 220.55, Note 4:

The branch-circuit load for a counter-mounted cooking unit and not more than two wall-mounted ovens, all supplied from a single branch circuit and located in the same room, shall be calculated by adding the nameplate rating of the individual appliances and treating this total as equivalent to one range.

NEC 220.55 and Table 220.55 also includes some derating (or more accurately, permissive up-rating) that overrides the normal "125% for continuous" derate found in NEC 210.19(A)(1). It appears to be permissive. Speaking of 210.19, ThreePhaseEel points out

210.19(A)(3) Household Ranges and Cooking Appliances. Branch- circuit conductors supplying household ranges, wall-mounted ovens, counter-mounted cooking units, and other household cooking appliances shall have an ampacity not less than the rating of the branch circuit and not less than the maximum load to be served. For ranges of 8 3/4 kW or more rating, the minimum branch-circuit rating shall be 40 amperes.

Exception No. 1: Conductors tapped from a 50-ampere branch circuit supplying electric ranges, wall-mounted electric ovens, and counter-mounted electric cooking units shall have an ampacity of not less than 20 amperes and shall be sufficient for the load to be served. These tap conductors include any conductors that are a part of the leads supplied with the appliance that are smaller than the branch-circuit conductors. The taps shall not be longer than necessary for servicing the appliance.

Exception No. 2: The neutral conductor of a 3-wire branch circuit supplying a household electric range, a wall-mounted oven, or a counter-mounted cooking unit shall be permitted to be smaller than the ungrounded conductors where the maximum demand of a range of 8¾-kW or more rating has been calculated according to Column C of Table 220.55, but such conductor shall have an ampacity of not less than 70 percent of the branch-circuit rating and shall not be smaller than 10 AWG.

This also overrides the 125% rating by saying a 40A breaker can definitely supply two 20A ovens, and saying a 8.75KW-9.6KW oven is allowed on a 40A circuit.